# Full text of "Pure Logic: Or, The Logic of Quality Apart from Quantity; with Remarks on ..."

## See other formats

This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project to make the world's books discoverable online. It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover. Marks, notations and other marginalia present in the original volume will appear in this file - a reminder of this book's long journey from the publisher to a library and finally to you. Usage guidelines Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing this resource, we have taken steps to prevent abuse by commercial parties, including placing technical restrictions on automated querying. We also ask that you: + Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for personal, non-commercial purposes. + Refrain from automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the use of public domain materials for these purposes and may be able to help. + Maintain attribution The Google "watermark" you see on each file is essential for informing people about this project and helping them find additional materials through Google Book Search. Please do not remove it. + Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner anywhere in the world. Copyright infringement liability can be quite severe. About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/ ^ WIpEjyER HN TK3C I ^ prjiiK 1 r f i lV LroH^.l ^/i Digitized by Google '*^§?!ifc.-.i-.>^' ..3 Digitized by Google Digitized by Googk ' PURE LOGIC THE LOGIC OF QUALITY. Digitized by Google LOKDOK rSIKTID BT BOWARD BTAKFORD 6 CHABIN6 CBOSS, 8. W. Digitized by Google PURE LOGIC OB THB LOGIC OF QUALITY APART FROM QUANTITY: wrrH REMABKS ON BOOLE'S SYSTEM AND ON THE RELATION OF LOGIC AND MATHEMATICS. W. STANLEY JEVONS, M.A. Logica est ars artinm et ecientia scientiarum. — Sootvs. LONDON: EDWARD STANFORD, 6 CHARING CROSS. 1864. Digitized by Google TVvl5~0H«t'l Digitized by Google CONTENTS. CHAPTER PAGE Introduction 1 I. OfTbbms 4 II. Of FBOFosrnoNS 8 in. Of DiBEOT IlTFBBBNCB 10 IV. Of Combination of Tbbhs 14 V. Of Skpabation of Tbbhs 22 VI. Of Plural Terms 25 Vn. Of Negative Propositions 29 VIII. Of Contbabt Tebms 31 IX. Of Contbaby Altebnatiyes . ... .36 X. Of Contbaby Tebms in Propositions . .39 XI. Of Indirect Infbbbnce 42 XII. Of Belation to Common Logic 53 XIII. Examples of the Method 58 XIV. COMPABISON WITH B00LE*S StSTBM . .69 XV. Ebmabks on Boole's System, and on the Belation of Logic and Mathematics ... 75 Digitized by Google Digitized by Google PURE LOGIC. INTRODUCTION. It is the purpose of this work to show that Logic assumes a new degree of simplicity, precision, generality, and power, when comparison in quality is treated apart from any reference to quantity. 1. It is familiarly known to logicians that a Extent and term must be considered with respect both to the «w^««^ of jnc(int7i€f, individual things it denotes, and the qualities, cir- cumstances, or attributes it connoteSy or implies as belonging to those things. The number of indi- viduals denoted forms the breadth or extent of the meaning of the term ; the qualities or attributes connoted form the depth, comprehension, or intent, of the meaning of the term. The extent and in- tent of meaning, however, are closely related, and in a reciprocal manner. The more numerous the qualities connoted by a term, the fewer in general the individuals which it can denote ; the one di- mension, so to speak, of the meaning being given, the other follows, and cannot be given or taken at will. Digitized by Google PURE LOGIC. Expression tLstmUy Separation necessary. Primary system. 2. Logicians have generally thought that a proposition must express the relations of extent and intent of the terms at one and the same time, and as regarded in the same light. The systems of logic deduced from such a view, when compared .with the system which may otherwise be had, seem to lack simplicity and generality. 3. It is here held that a proposition expresses the result of a comparison and judgment of the sameness or difference of meaning of terms, either in intent or extent of meaning* The judgment in the one dimension of meaning, however, is not independent of the judgment in the other dimen- sion. It is only then judgment and reasoning in one dimension which is properly expressed in a simple system. Judgment and reasoning in the other dimension will be and must be implied. It may be expressed in a numerical or quantitative system corresponding to the qualitative system, but its expression in the same system destroys! simplicity. I do not wish to express any opinion here as to the nature of a system of logic in extent, nor as to its precise connection vdth the pure system of logic of quality. 4. Keasoning in quality and quantity, in intent or extent of meaning, being considered apart, it seems obvious that the comparison of things in quality, with respect to all their points of same- ness and diflference, gives the primary and most general system of reasoning. It even seems likely that such a system must comprehend all possible and conceivable kinds of reasoning, since it treats Digitized by Google INTRODUCTION. 3 of any and every way in which things may be same or different. All reasoning is probably founded on the laws of sameness and difference which form the basis of the following system. 6. My present task, however, is to show that Present all and more than all the ordinary processes of logic may he combined in a system founded on comparison of quality only^ without reference to logical quantity. 6. Before proceeding I have to acknowledge Belation to that in a considerable degree this system is foimded -j. * on that of Prof. Boole, as stated in his admirable and highly original Mathematical Analysis of Logic* The forms of my system may, in fact, be reached by divesting his system of a mathe- matical dress, which, to say the least, is not essential to it. The system being restored to its proper simplicity, it may be inferred, not that Logic is a part of Mathematics, as is almost im- plied in Prof. Boole's writings, but that the Ma- thematics are rather derivatives of Logic. All the interesting analogies or samenesses of logical and mathematical reasoning which may be pointed out, are surely reversed by making Logic the dependent of Mathematics. * Investigation of the Laws of Thought. By George Boole, LL.D. London, 1854. Frequent reference will be made to this work in the following pages. B 2 Digitized by Google PURE LOGIC. CHAPTER I. Of things and their Meaning of name. OF TERMS. 7. Pure logic arises from a comparison of things as to their sameness or difference in any guality or circumstance whatever. , i In discourse we refer to things by the aid of marks, names, or terms, which are also, as it were, the handles by which the mind grasps and retains its thoughts about things. Thus correct thought about things becomes in ^scourse the correct use of names. Logic, while treating only of names, ascertaining the relations of sameness and difference of their meanings, treats indirectly, ^ , as alone it can, of the samenesses and differences^V^> of thmgs. \ 8. A term taken in intent has for its meaning '. the whole infinite series of qualities and circum^ ^' stances which a thing possesses. Of these qualities or circimistances some may be known and form the description or definition of the meaning ; the infinite remainder are vmknown. Among the circumstances, indeed, of a thing, is the fact of its being denoted by a given name, but we may speak of a thing, of which only the name is known, as having a name of unknoivn meaning. The meaning of every name, then, is either un- Digitized by Google TERMS. 5 known or more or leas known. But we may speak of a term that is more or leas known as being simply kjwwn. 9. Among the qualities and circumstances of a Qualities thing is to be counted everything that may be said ^^Jj^/** of it, affirmatively or negatively. Any possible quality or circumstance that can be thought of either does or does not apply to any given thing, and therefore forms, either affirmatively or nega- tively, a quality or circumstance of the thing. Concerning anything, then, there may be an in- finit|@ number of statements made, or qualities predicated. 10. When we assign a name to a thing, with Bdation of knowledge of, and regard to, certain of its qualities ^''^^*'^*' or circumstances, that name is equally the name of anything el^e of exactly the same known quali- ties and circumstances. For there is nothing in the name to determine it to the one thing rather than the other. Any name, then, must be the name in extent of anything, and of all things agreeing in the qualities or circumstances which form its known meaning in intent, and in this system. 11. Though it is well to point out that all our Present names or terms bear a universal quantity when. ^"^^^' regarded in extent, it must be understood, and constantly borne in mind, that ftirther reference to the meaning of a term in extent or quantity of individuals, is excluded in these pages. jThe primary and only present meaning of a name or term is a certain set of qualities, attributes^ * properties, or circumstances^ of a thing unknown or partly known. Digitized by Google PURE LOGIC. Term defined. Generality of our terms. Proper names. Condition of same- ness of meaning. 12. Term will be used to mean name, or any combination of names and words describing the qualities and circumstances of a thing. 13. The terms of this system may be made to express any combination of samenesses and diffe- rences in quality, kind, attribute, circumstance, number, magnitude, degree, quantity, opposition, or distance in time or space. A term may thus represent the qualities of a thing or person in all the complexity of real existence, so well and fully defined that we cannot suppose there are, or are likely to be, two things the same in so many circumstances. Such a term would correspond to the singular, proper, non-attrihutive, or non-con- notative names of the old logic. Such names are accordingly by no means excluded from this sys- tem ; and it is here held that the old distinction oi connotative and nxm-connotative names is wholly erroneous and unfounded. If there is any dis- tinction to be drawn, it is that singular, proper, or so-called non-connotative terms, are more full of connotation or meaning in intent or quality than others, instead of being devoid of such meaning. 14. As logic only considers the relations of meaning of terms, as expressed within a piece of reasoning, the special meaning of any term is of no account, provided that the same term have the same meaning throughout any one piece of reasoning. Thus, instead of the nouns and adjectives, to each of which a special meaning is assigned in common discourse, we shall use certain letters, A, B, C, D, . . . • U, V . . . each standing Digitized by Google TERMS. 7 for a special term, or a definite meaning, and for any term or meaning^ always under the above condition. 16. Our terms, A, B, C, like the Term terms of common discourse, may be either known' u^f^jcnown, or imknown in meaning. It is the work of logic to show what relations of sameness and difference " between unknown and known terms may make the unknown terms known. Were it not to explain ignotum per ignotiv^s^ we might say that logic is the algebra of kind or quality^ the calculus of known and unknown quali^ tieSy as algebra (more strictly speaking universal arithmetic, which does not recognise essentially negative quantities) is the calculus of known and imknown quantities. 16. Let it be borne in mind that the letters A, B, Symbols of C, &c., as well as the marks +, 0, and =, after- ^fieaning, wards to be introduced, are in no way mysterious symbols. The term A, for instaace, is merely a convenient abbreviation for any ordinary term of language, or set of terms, such as Bed, or the Lords Commissioners for executing the office of the Lord High Admiral of England, Again, + is merely a mark substituted for the sake of clearness, for the conjunctions and, either, or, &c., of common language. The mark = is merely the copula is, or is same as, or some equivalent. The meaning of 0, whatever it ex- actly be, may also be expressed in words. There is consequently nothing more symbolic or myste- rious in this system than in conamon language. Digitized by Google PURE LOGIC. CHAPTER n. OP PROPOSITIONS. Proposi' tion dc' Affirma- tive, negative. Its purpose. Truth and falsity. Notation of affirma" Uve propo' sition. 17. A proposition is a statement of the samenesst or difference of meaning of two terms, that is, of the sameness or difference of the qualities and circum- stances connoted by each term. 18. According as a proposition states sameness or difference^ it is called affirmative or negative, 19. It is the purpose or use of a proposition to make known the meaning of a term that is otherwise unknown. 20. A proposition is said to be true when the meanings of its terms are same or different, as stated; otherwise it is called false or untrue. As logic deals with things only through terms, it cannot ascertain whether a proposition is true or false, but only whether two or more propositions are or are not true together, imder the condition of meaning of terms (§ 14). 21. We denote by the copula is, or by the mark =, the sameness of meaning of the terms on the two sides of a proposition. For the present we shall speak only of affirma- tive propositions, which are of superior importance ; and when not otherwise specified, proposition maj be taken to mean affirmative proposition. Digitized by Google PROPOSITIONS. 9 22. A proposition is simply convertible. The Conversion propositions A=B and B=A axe the same ^fP^^^^P*^' statement; either of the terms A and B is the same in meaning as the other, nndistinguiBhable except in name. This simple conversion comprehends both the simple conversion, and conversio per accidens of the school logic. 23. One proposition and one known term may One term make known one unknown term. known _ . -w^ « , ■« 1 . from one From A=B, so far as we know B, that is, proposi- know its meaning, we can learn A ; so ^ as we ^*^^- know A, we can learn B. We thus know samely of both sides of a propo- sition whatever we know of either. The same might be said of uncertain or obscure knowledge. 24. A proposition between any two terms of Useless which the meanings are otherwise known as same ^^ *jf^^f" or different, is useless. For it cannot serve the sitions purpose of a proposition (§ 19). Such is any "^^w^^^^- proposition between a term and itself, as A=A, B=:B (§ 14). These useless propositions are called Identical, They state the condition of all reasoning, but we know it without the statement. A proposition repeated, or a converted propo- sition (§ 22), is also useless, except for the mere convenience of memory, or ready apprehension. Digitized by Google 10 PURE LOGIC. CHAPTER III. OF DIRECT INFERENCE. Law of 25. It is in the nature of thought and things, sameness. ^^^ things which are same as the same thing are the same as each other. More briefly — Same as same are same. Hence the first law of logic — that terms which are same in,meaning as the same term^ are the same in meaning as each other. This law, it is obvious, is analogous to Euclid's first aadom, or common notion, that things which are equal to the same thing, are equal to each other. Things are called equal which are same in magni- tude, but what is true of such sameness, is also true of sameness in any way in which things may be same or different. Euclid's geometrical law is but one case of the general law. Meaning of 26. Logic proceeds by laws, and is bound by kiws of them. For logic must treat names as thought treats things. And the laws of logic state certain samenesses or uniformities in our ways of thinking, and are of self-evident truth. Direct in- 27. When two affirmative propositions are same shown and *^* ^^^ member of each, the other members may be defined. stated to be same. From A=B, B=C, which are the same Digitized by Google DIRECT INFERENCE, 11 in the member B, we may form the new propo- sition A=C. For A and C being each stated to be the same as B, may by the law of sameness be stated to be the same as each other. A proposition got by the Law of Sameness is said to be got by direct inference^ and is called a direct inferentj or, in common language, a direct inference, 28. Propositions from wnich an inference is Premise drawn are called premises^ and are given or taken '^^^ ' as the basis of reasoning. Logic is not concerned with the truth or falsity of premises or inference, except as regards the truth or falsity of one with the other. (§§ 20, 37.) 29. An expression for a term consists of any Expression other term which by premises we know to be the "^^ ' same in meaning with that term. 30. In inferring a new proposition from two Elimina- premises we are said to eliminate or remove j^^^^, the member which is the same in the two pre- mises. From two premises we may eliminate only one term, and infer one new proposition. By saying that we may, it is not meant that we always can, 31. Propositions are said to be related to each Related other which have a same or common member, or fjj^ which are so related to other propositions so re- d^md. lated ; and so on. In other words, any two propositions are related which form part of a series or chain of propositions, in which each proposition is related to the adjoin- ing ones or one. Digitized by Google 12 PURE LOGIC. Belated terms de* Use of syUogism, Series oj premises. NuTnher of terms and related, 'premises Irrelevant terms and premises. 32. Terms are said to be related which occirf in one same, or in any related propositions. 33. From two related premises and one known term we may learn two unknown terms, and not more. From A=B and B=C, we learn any two of A, B, C, when the third is known. 34. From any series of related premises, and one known term we may learn as many unknoivn terms as there are premises. Thus, from A = B = C=D=E=F, we may learn any five terms when the sixth is known. For each useful propo- sition may render one unknown term known (§ 19). Between each two adjoining premises one term may be eliminated, becoming known in one premise, and rendering another term known in the other. There must at last remain a single proposition containing two terms, each of which occurs only in one premise. 35. The number of related premises must be one less than the number of different terms. If it be still less, the propositions cannot be all related ; if it be greater, some of the premises must be useless, because they must lie between terms otherwise known to be same by inference. It will be remarked that systems of mathe- matical propositions or equations with known and imknown qualities are perfectly analogous in their properties to logical propositions. 36. When a related premise contains a term or member not relevant to the purpose of the reasoning, this term is eliminated by neglecting the premise ; and for every such premise neglected Digitized by Google DIRECT INFERENCE. 13 a term is eliminated. In regard both to related and unrelated premises and their terms, the neglect of all irrelevant terms and premises may be con- sidered a process of elimination which accompanies all thought. 37. Inference is judgment of judgments, and Science of ascertains the sameness of samenesses. Science, When in comparing A with B, and the same B with C, we judge that A=B and B=C, we ob- tain sciencej or reasoned knowledge of things, as distinguished from the mere knowledge of sense or feeling. But when we judge the judgments A=:B=C to be the same, as regards A and C^ with the judgment A=C, we obtain Science of Science, Here is the true province of logic, long called Scientia Scientiarum. Hence it is that logic is concerned not with the truth of propositions per S6 (§ 20), but only with the truth of one as depending on others. SCIENCB OF SCIBKCB {A«H=C} = {A«C} Beasonino SciENCB A=B B«C Judgment Things ABC Apprehension 38. Here we find the clear meaning of the Form and distinction of form and matter of thought. matter. Sameness of Samenesses » Form ) \ Sameness of things - j ^^^^ \ [ Of thought Things « Matter)] Digitized by Google 14 PURE LOGIC. ings, CHAPTER IV. OF COMBINATION OF. TERMS. Addition 39. In discourse, when several names are %ia8^^^' placed together side by side, the meaning of the joint term is sometimes the sum of the meanings of the separate terms.* So in our system, when two or more terms are placed together^ the joint term must have as its meaning the sum of the meanings of the separate terms. These must be thought of together and in one. * I shall here consider only the cases of combina- tion in which the combined term means the added meaU' ings of the separate terms. The same forms of reasoning apply, as I believe, mutatis mutandis^ to any cases of combination imder some such wider law as this — Same parts samdy related make same wholes. Only by some such extension can logic be made to embrace the major part of all ordinary reasoning, which has never yet been embraced by it, save so far as this may have been done in some of Professor De Morgan's latest writings. But to show how such an extension may be grafted on to my system must be reserved for a future opportunity. In most relations it is obvious that the order of terms in relation is no longer indifferent. (§ 41.) Concerning some inferences by combination, see Thom- son's Outlines^ §§ 87, 88. Digitized by Google COMBINATION. 15 40. Any terms placed together will be said to Combina- form with respect to any of those separate terms 1^^ a combination or combined term. With respect to all other terms they may be called simply a term. For it must be remembered that any single term, A, B, C, &c., is not more single in meaning than a combination. 41. The meaning of a combination of terms is Order of the same in whatever order the terms be combined. J?^ indif- Thus, AB = BA; ABCD = BACD = ferent. DCAB, and so on. For the order of the terms can at most affect only the order in which we think of them, and in things themselves there is no such order of qualities and circumstances. (Boole, p. 30.) 42. A combination of a term with itself is the Law of same m meaning with the term alone, '^ ^ Thus AA = A, AAA = A, and so on. Also, a combination of terms is not altered by combination with the whole or any part of itself. Thus ABCD = ABCD . BCD=A . BB . CC . DD =ABCD, since BB = B, CC = C, DT> = D. The coalescence of same terms in combination must be constantly before the reader's mind. This important and self-evident law of logic was first brought into proper notice by Prof. Boole (p. 32), who remarks : * To say " good, good," in relation to any subject, though a cum- brous and useless pleonasm, is the same as to say " good." ' Professor Boole gave to this law the name Law of Duality. But as this name, on the one hand is not peculiarly, adapted to express the Digitized by Google 16 PURE LOGIC. Degree of quality. Law of same parts and wholes. Inference by com- bination. general feet AAAAA = A, and is pecu- liarly adapted to express the fact A = AB + Ah (§ 99), I have yentured to transfer the name, and substitute a new one. 43. In the terms as used under the above law there is no reference to degree of quality. When required, each degree of quality may be treated in a separate term, containing as part of its mean- ing every less degree of the quality. Two or more degrees of a same quality in logical com- bination therefore produce the greatest of those degrees. 44. ■ It is in the nature of thought and things that ie;Aen same qualities are joined to same qualities the wholes are same. Hence the law of logic — Same terms combined with same terms give same combined terms. Thus, since A = A and B = B, therefore AB = BA = AB. This self-evident law is a more general case of Euclid's second axiom. It may, perhaps, be most briefly stated as follows i^-Same parts make same wholes, 48. Same terms being combined vnth both mem- bers of a premise, the combinations may be stated as same in a new proposition which will be true with the premise. For what is true of terms obviously the same, as A, A, or B, B, must also be true of terms known to be the same in meaning by a premise. Thus, from A =s B we may infer AC = BC by combining C with each of A and B. Digitized by Google COMBINATION. 17 As the number of possible terms which may be combined with the terms of a premise is infinite, there may be drawn from any premise an infinite number of inferences by combination. 46. Inferences which may be drawn by com- Combiner bining the members of two or more premises need ^^.^fP^' not be considered here. 47. A proposition inferred by combination Gmend (§ 45) will be true with its premise, whatever be J^^ ^^ the term or terms used for combination. When f&rences. terms of specific meaning, indeed, are selected at random, it will usually happen that the combina- tions of the inference are imheard-of, absurd, and useless. This does not affect the truth of the in- ferred proposition, which only asserts that the meaning of the one combination, whatever it be, is the same as the meaning of the other. - 48. In our daily use of specific terms, we con- Tacit rela- stantly use each under the restriction of a number ^t^, J^" of premises so well known to all persons that it is needless to express them. Terms joined not in accordance with these tacit relations make non- sense. For instance, the impassable difference of matter and mind renders it nonsense to join the name of any material with that of any mental at- tribute, except in a merely metaphorical sense. In order, then, that our inferences should always be intelligible and useful, we should require the expression of all tacit premises connected with terms of specific meaning. It is only the several branches of science, however, that can undertake the necessary investigations in detail. Our infer- ence remains true, however complicated be the Digitized by Google 18 PURE LOGIC. Formation of common term. Substitu- tion cU' fined. relations of sameness and difierence of the terms introduced. But it is inference from premises which are stated^ not from those which migM he or ought to he stated, • 49. When premises contain terms only par- tially the same, the combination of each with the part that is different in the other will produce a term completely the same in each. Such premises may be considered as related. (§31.) Thus, in A = C and B = CD, the terms C and CD are only partially the same. But the combination of D with A = C gives AD = CD, having one member completely the same as one member of B = CD. Hence we may infer AD = CD = B (§ 26), and eliminate the term C, which was common in the premises : thus, AD = B. Again, to eliminate B from the premises A = BC and E = BD, combine D with each side of the first, and C with each side of the second. Hence, AD = BCD = CE, or AD = CE, in which B does not appear. 60. From premises which have no term in common, this process will only give us the in- ferences which might be had (§ 46) by the direct combination of the respective terms of the pre- mises. Thus, A = B and C = D give AD = BD, and BC = BD, whence AD = BC. And we might similarly get AC = BD. 61. The following process may be called suh- stitution, and will be seen to give the same inference as the two processes of forming a common term (§§49, 27), and then eliminating it. Digitized by Google COMBINATION. 19 F&r any temtj or part-term, in one premise, may } » / be suhatituted its expression (§ 29) in other terms. ^ * In short, the two members of any premise may . , be used indifferently, one in place of the other, wherever either occurs. Thus, if A=BCD and BC=E, we may in the former premise substitute for BC its expression E, getting A=DE. The fiill process of inference consists in combining D with both sides of BG=E, and eliminating the complete common term BCD thus obtained, so that A=BCD=DE. 52. We may substitute for any part of one intrinsic member of a proposition the whole of the other. elindna- Thus, in A=BCD, we may substitute for any one of B, C, D, BC, BD and CD, parts of BCD the one member, the whole, A, of the other member, inferring the new propositions — A=ACD A=ABD A=ABC A=AD A=AB A=AC. The validity of this process depends on the ) Laws of Simplicity (§ 42), and of Part and / Whole (§ 44), as is seen by combining each mem- ber of the premise with itself. Thus, from A=BCD we have A.A=BCD.BCD=BCD.D =AD by coalescence of same terms, and sub- stitution for BCD of its expression A. The new proposition thus inferred will have one of its sides pleonastic, that is, with some part ■ of its meaning repeated. But it is obvious that we cannot, as a general rule, substitute for part of one side less than the whole of the other, because Digitized by Google 20 PURE LOGIC. we cannot from the premise alone know that the meaning of the part-term removed is quite sup- plied in the part of the other member put for it. The above process may be called intrinsic eli- minatiouy to distinguish it from the former process of elimination between two premises, which may be called extrinsic elimination, and is seen to be that case of intrinsic elimination in which we substitute for the whole of one side the whole of the other. In a single premise, intrinsic elimina- tion of a whole member would give only an identical and useless result. Intrinsic elimination gives no new knowledge, but is of constant use in striking out or abstract- ing terms concerning which we do not desire knowledge, and which are therefore worse than useless in our results. Professor Boole's system of elimination (p. 99), is, I believe, equivalent to the above, though the correspondence may not at first sight be apparent. Failure of 53. A term cannot be intrinsically eliminated ^ti^^^' which occurs in both members of a proposition. The presence of such part-term may be called a condition of the sameness of the remainder of the terms. 54. Terms are said to be samely related in a premise when their interchange does not alter the premise. Thus, B and C are samely related in A=BC, because the premise is the same A=GB (§ 41) after their interchange. But A and B are not samely related, because their interchange altera the premise into B=:AC. tion. lated terms, Digitized by Google COMBINATION. 21 In A=BCDE . . . any two of B, C, D .... are samely related and may be interchanged. 55. Of samely related terms, an expression Inference for the one is the same as the expression for an- ^^^^^ other after the two terms in question have been interchanged. 66. When several terms are samely related, Chncem- we obtain the expressions concerning the rest »^*^^^^- from the expression for any one by successively changing each term into the next when the terms are kept in some fixed order. It is evident that we may always interchange the terms in any part of a problem, provided we do so throughout the problem (§ 14). And in those cases in which the premises remain un- changed thereby, we evidently get several infer- ences from the same premises. This method of interchanges is familiar to mathematicians. 57. It will be obvious that a mathematical Mathema- term or quantity of several factors is strictly ana- f*^ ^^^" logons in its laws to a logical combined term, excluding the Law of Simplicity, Digitized by Google 22 PURE LOGIC. CHAPTER V. OF SEPARATION OF TERMS. Law of 58. It is in the nature of thought and things that ^andmrts '^^^fi'^^^ *^^^ ^^^* of qualities same qualities are taken, the remaining sets are the same ; or, more briefly — Same parts from same wholes leave same parts. Hence the logical law: — When from same com- binations of terms same terms are taken, the re- maining terms are the same. This is the converse of the Law of Same Parts and Wholes (§44), and is equally self-evident with it. But it is not equally useftd with it ; and in Pure Logic, in &ct, is of no use at all. The removal of terTtis with their known meanings is not equally possible with their combination, and in useful logical premises, is not possible at all. For, in a useful premise (§ 19), a part at least- of one member must be unknown, and this part may or may not contain the part we desire to remove. Even supposing then that a term occurs on either side of a premise, we cannot remove it from the known side, because we cannot know whether or not we can remove it from the imknown or par- tially known side. Thus in AB^AC, suppose A and G known, Digitized by Google SEPARATION. 23 and B iinknown. We cannot infer B=C, because B may contain part or the whole of the known meaning of A, in addition to the known meaning of C, by the Law of Simplicity (§ 42), and in leaving B, we do not remove A from one member of the premise. 69. The logic of known and unknown terms, ConvpUu it has been said (§ 15), is analogous to the calculus J^^^^y jv of known and unknown numbers. mathema- So, a logic in which all terms were known ^*^*' would have an analogue in common Arithmetic, a calculus in which all the numbers employed are known. Combination of terms has an analogue in multiplication of numbers, and separation of terms in division of numbers. As in logic combination is tmrestricted, so in calculus is multiplication. As in logic of known terms only, separation of terms is unrestricted, so in a calculus of known numbers only, division is unrestricted. But, as in logic of known and unknown terms separation is restricted^ so in calculus of known and unknown numbers division is restricted. 60. It is well known that, in like manner, we Restriction cannot divide both sides of an equation by an ^Z^*^'***^- unknown fector, and assert the resulting equation to be necessarily true, because the unknown fector may be = 0. Thus, from xy = xz, we cannot remove x, and assert y =^ z, because if X happen to be = 0, the equation xy = xz is true, whatever finite numbers be the meanings of ^and z. The correspondence is thus shown : — Digitized by Google 24 PURE LOGIC. Parts not known fr&in the whole. Number of terms and pre- mises. Logical Fbopostttons. TerTTis known admit Combination Separation (unless either diyidend contain divisor) Terms unknown admit Combination but do not admit jSeparation Mathematical Equatioits. Numbers known admit Multiplication Division (unless diyisor =s 0) Numbers unknown admit Multiplication but do not admit DiTiaion The above, analogies did not escape the notice of Professor Boole (pp. 36-37), and I am therefore at a loss to understand on what ground he asserts that there is a breach in the correspondence of the laws of logic and mathematics. 61. From the meaning of a whole term we can- not learn the meaning of a part. In A=BC, if we know A we learn BC as a whole; but we do not thence learn the parts B, C, separately. For of the qualities in A any part may be in B, and any part in C, including any part of those in B, by the Law of Simplicity (§ 42). It is only necessary that every quality in A shaU be either in B or in C. Even if we know one of B and C, we only learn of the other that it must contain any quality of A not in the first We here meet the imperfection of an inverse process. 62. With reference to the relation between the number of premises, and the numbers of known and unknown terms (§§ 33-35), we must treat as separate terms any which occur separate in pre- mises, although they may also occur in combina- tion. Otherwise, we always treat any whole combination as a single term. Digitized by Google PL URAL TERMS. 25 CHAPTER VI. OF PLURAL TERMS. 63. A plural term has one of several meanings^ Terms of hut it is not known which, ^n^,\,^o Tneanmgs. Thus B or C is a plural term, or term of ;iiany meanings, for its meaning is either that of B or that of C, but it is not known which. A term not in form plural, may be distinguished as single ; such is A. 64. The separate terms expressing the several Alternative possible meanings of a plural term are called aZ- temativesy and are to be joined together by the sign + placed between each two adjoining terms. All that has been said of single terms applies to plural terms, mutatis mutandis. 66. The meaning of a plural term is the same Order of whatever be the order of the alternatives. ^ws^' Either B or G is the same in meaning as either C or B, that is, B + C=C + B. For the order in which we think of the possible qualities of a thing cannot alter those qualities, and the order must not convey any intimation that one meaning is more probable than another. 66. A term is combined with a plural term by Combiner combining it with each of its alternatives. ^^^ For what is A and either B or C, if it is B, term. Digitized by Google 26 PURE LOGIC. Use of brackets. Combina- tion of plural terms. Law of unity. Jltums terms. is AB ; if it is C, is AC, and it is therefore either AB or AC. 67. Let a plural term enclosed in brackets ( ), and placed beside another term, mean that it is combined with it, as one single term is with another : Thus A (B -f C) = AB + AC. 68. One plural term is combined with another by combining each alternative of the one separately with each of the other. Each combined alter- native may then be combined with each alternative of a third plural term, and so on : Thus(DH-E)(B + C)=B(D+E) +C(D + E) =BD + BE + CD-|-CE. 69. It is in the nature of thought and things that same alternatives are together same in meaning^ as any one taken singly. Thus, what is the same as A or A is the same as A, a self-evident truth. A+A=A A-|-A+A=A A+A+B=A+B This law is correlative to the Law of Simplicity, (§ 39), and is perhaps of equal importance and frequent uee. It was^not recognised by Professor Boole, when laying down the principles of his 70. In a plural term, any alternative may be re- moved, of which a part forms another alternative. Thus the term either B or BC is the same in meaning with B alone, or B + BC=B. For it is a self-evident truth (§99) that B standing alone is either the same as BC, or as B not-C. Thus B+BCs=B not'C+BC+BC =B no^C+BC=B. Digitized by Google LAW OF UNITY. 27 71. A plural term obeys the Law of Simplicity Plural For let A=B -h C ; then — single AA=(B+C)(B-hC). '^^'^• AA=BB-fBC+BC+CC (§68). A= B+BC+C (§42). A= B + C (§70). A plural term obeys the Law of Unity (§ 69) : A+A=B + C+Bh-C = B-i-C. 72. For any alternative or part of an alternative Subsiiitt- may be substituted (§ 51) its expression in other ^^^^^ terms : tenm. Thus, if A=B+CD and D=E, substitute, getting A=B + CE. 78. A plural term may be substituted like a Suhstitu- singleterm for any term, single or plural, of which J^^.^ it is the expression. When in combination, the terTna. several alternatives must be separately combined (§§ 66, 68). Conversely, for a plural term may be substituted its expression in a single term : Thus, if A=BC and C=D4-E, for C sub- stitute D+E, and A=B (D+E)=BD4-BE. Or from the premises A=BD+BE= B(D+E) and C=D-hE, we might by substitu- tion get back to A=BC. 74. A plural term is known when each of its Plural alternatives is known. ^f^ known. Thus, in A=B -|- C, A is known when the meanings of each of B and C are known. But of course from knowing a single meaning of A, we cannot learn either or both of B and C. Digitized by Google 28 PURE LOGIC. Ntmher of 76. With reference to the relation between the terms and number of premises, and the nmnbers of known premises. _ _ ^ ,«.«.«« *»,-v and imknown terms (§§ 33-35), we must treat as a separate term each alternative of a plural term. A proposition with a plural term thus corre- sponds to an equation with several unknown quantities. Plural and 76. As plural terms obey the laws of single ^i^gl^ terms, and a term single in form may be plural in meamng, it will not be necessary for the future to distinguish plural arid single termSy any more than it has been to distinguish combined and simple terms. There is some danger of misconception con- cerning plural terms. Though a plural term has one of several meanings, it cannot bear in this system more than one at the same time, so to speak. Hence it still remains a unit^ the name of a single set of qualities, one of several sets, but it is not known which. The whole of this system in short is unitary ^ and involves the same remark- able analogies to a calculus of unity and which have been brought forward so explicitly in Professor Boole's system. Digitized by Google NEGATIVE PROPOSITIONS. 29 CHAPTER Vn. OF NEGATIVE PROPOSITIONS. Terms may also be known and stated as differ- ing, or not being the same in meaning. 77. It is in the nature of thought and things Law of that a thing which differs frcm another differs ^€^^^^* from everything the same as that other. More briefly stated — Same as different are different. Hence in logic — A term which differs from another term in meaning differs from every term which is the same as that other. If A is not the same as B, which is the same as C, then A is not the same as C. The infe- ren^ce arises in the sameness of B and C, allowing us to substitute one for the other. Hence we learn nothing of the sameness or difference of any two terms, D and E, each of which differs from a third, F ; for D and E may each have any of an indefinite variety of meanings, and each may yet differ from F. (§152.) 78. Hence a chain of related premises between Neaative any of which inferences can be drawn, must not ^^fi^^<^^' contain more than a single negative premise. Digitized by Google 30 PURE LOGIC. Conver- sion. Law of different parts and wholes. Law of different wholes. Also any inference in whidi a negative premise is concerned must be a negative inference. 79. A negative proposition is simply convertible. For A is not the same as B, is the same state- ment as JB is not the same as A. 80. When same terms are combined with diffe- rent termSy the wholes may be different If A differs from B, then AC differs from BC, provided, however, that the difference of A and B does not consist in any part of C. 81. When from different wholes same parts are taken, the remainders are different. This is equally self-evident with the preceding converse. It is unnecessary further to consider negative propositions, because their inferences may be ob- tained by use of aflirmative propositions. Digitized by Google CONTRARY TERMS. 31 CHAPTER VIII. OF CJONTRARY TERMS. 82. The known meaning of a negative term is Negative the absence of the quality^ or set of qualities^ which ^^^' forms the known meaning of a certain other, itspo' sitive term. Thus not'A is the negative term signifying the absence of the quality or set of qualities A, If the known meaning of A be only a single quality, not-A means its absence ; but if A mean several qualities, not-A means the absence of any one or more. Thus, if ^=5.(7 not'A = B not'C + not-B.C + not-B not-C. 83. The negative of a negative term is the cor^ Negative of responding positive term. negative. What is not-not'A is A. 84. Since the relation of a positive to a negative Simple term is the same as the relation of a negative to a <^^^^'^ positive, let each be called the simple contrary fined, term of the other. 85. For convenience let not'A be written a. Notation. Then any large and its small letter denote a pair of simple contraries; and not-a is A. Also, the contrary of BC (§ 82) is Be -^ bG + be. Digitized by Google 32 "PURE LOGIC. Laws ' Involve * defined. Contrary of plural term. Contrary combina- tions. Law of contradiC' tion. ContradiC' tory term defined. which expresses the absence of one or more of B andC. 86. All that has been said of a term applies samely to one as to the other of a pair of contraries. Thus, a obeys the several laws : C=:D J aa^=^a a-^a^a p___ jv and so on. 87. Let a combined term or a proposition be said to involve a term when it contains either that term or its contrary. 88. The contrary of a plural term is a term containing a contrary of each alternative. Thus the contrary of A + B + C is ahc. If any alternative has more than one contrary, for each there will be a contrary alternative. Thus, A + BC has the plural contrary aBc +ahC-\- ahc. 89. Any combined tenp which contains the simple contrary of another term may be called a contrary, or contrary combination of this, or of any combination containing this. Thus, any combined term containing A is a contrary of any term containing a, and it will seldom be necessary to distinguish by name simple contraries, such as A and a from contraries, or contrary conpibinations in general, which merely contain A or a. (See, however, §§ 99, 100.) 90. It is iu the nature of thought and things that a thing cannot both have and not have the same quality, 9L Hence a term which means a collection of qualities in which the same quality both is and is not, cannot mean the qualities of anything which is or ever will be known. Digitized by Google CONTRARY TERMS. S3 Such a term then has tw? meaning, that is to say, no possible, useful, or thinkable meaning ; but it may be said to mean nothing. Let it be called a self-contradictory, or, for sake of brevity, a con^ tradictory term, 92. Let us denote by the term or mark , l^se of 0. combined with any term, that this is contradic- tory, and thus excluded from thought. Then Aa=Aa.O, Bi=B6.0, and so on. For brevity we may write Aa=0, Bft=0. Such propositions are tacit premises of aU reasoning. Any two contrary terms in combination give a contradictory term. 98. Ai^ term being combined with a contra- Combina- dictoiy, the whole is contradictory. ^JZt^dL For the whole then means a collectibn of tory. qualities which does and does not contain some same quality, and is therefore by definition a con- tradictory. Thus, if A=B6 =BJ.O AC=BftC=B&C.O. 94. The term 0, meaning excluded from thought, Term 0. obeys the laws of terms. 0.0=0 + 0=0, otherwise expressed : — ^What is excluded and ex- cluded is excluded — What is excluded or excluded is excluded. 96. Any term not known to he contradictory Condition must he taken as not contradictory, ' ofnon-con- . , 1 ,. . tradiction. Any term known to be contradictory is excluded from notice, and any term concerning which we are desiring knowledge must therefore be assumed not contradictory. Digitized by Google 34 PURE LOGIC. ContradiC' tory alter- natives. Mimina^ Hon of con- tradictory. Elimina- tion of al- ternatives. 96. In a plural term of which not all the alter- natives are contradictory^ the contradictory alter* native or alternatives must he excluded from notice. If for instance A=0+B, we may infer A=B, because A if it be is excluded ; and if it be such as we can desire knowledge of, it must be the other alternative B. 97. No contradictory term is to he eliminated in direct inference. For all we can require to know of a contradic- tory term is that it is contradictory, and elimina- tion of a contradictory term would prevent rather than give such knowledge. Thus if A=Cc . 0, B=Cc . 0, all that we can require to know of A and B is known from these premises, and cannot be known from the inference A=B got by eliminating the contradic- tory Cc. 0. So, if A=B=C=D=E=F=G^.O, the only useful inferences are those showing each of A, B, C, D, E, F, to be contradictory. So, also, obviously, of intrinsic elimination. It may be said, in fact, that contradiction super- sedes all other elimination by itself eliminating all contradictory terms from fttrther notice. 98. An alternative is eliminated when its plural term is combined with a contrary of that alterna- tive. Thus, the alternative Ab is removed from the plural term AB + Ab when combined with B. (AB-H Ab)B = AB H- ABb = AB + = AB Digitized by Google CONTRARY TERMS. 35 Let C=AB+Aft+aB + aJ Then AC=AB+A5 ABC=AB BC=AB4-aB AJbC ^Ah aC=aB +ah aBC =aB 5C=:Aft +a6 aftC =aft. The term thus combmed with each side cannot be eliminated intrinsically (§ 53), and remains a condition of the rejection of the other alternative. It is by this rejection of alternatives that the extent or -width of the meaning of a term is reduced, as its intent of known meaning is in- creased, by combination (§^ 1). For every general term, in addition to its known meaning, may be assimied to have an indefinite multitude of unknown alternatives. In combination with a new terra many of these will probably become contradic- tory. D 2 Digitized by Google 86 PURE LOGIC. CHAPTER IX. OP CONTRARY ALTERNATIVES. Law of duality. Apparent exceptions. 99. It is in the nature of thought and things that a thing is either the same or not the same as another thing. Otherwise — A set of qualities either does or does not contain ' a certain quality. Hence, in logic, a term must contain the meaning of one of any pair of simple contrary terms. Thus : — A term is not altered in meaning by combination with any simple contrary terms as alternatives. A=A(B + *) = AB+A5. For if A has meanings containing only B, then A 5 is contradictory, and A=AB + 0=AB. If A has meanings containing only b, then AB=0 and A=0 + A6=Aft. If A has meanings of which some contain B and some b, the law is still true. This Law of Duality is not the same as Pro- fessor Boole's law of duality. (See § 42.) 100. Some apparent exceptions may occur to this law. For instance, let A= virtue, B=:black, and 5= not-black. Then the statement Virtue is either black or not-bl<zck, seems true according to the above law, and yet absurd. This Digitized by Google LAW OF DUALITY. 37, arises from B and h not being simple contraries ; for B may be decomposed into black-coloured — say BC, and b into not-black-coloured, or not black and not coloured, or bC + be. Now, virtue is really not coloured at all, or is Abe, and, therefore, neither BC nor bC Here, again, we must observe that the combination Be is contradictory from the tacit premise black is a colour (§ 48). Other apparent inconsistencies may be similarly explained. Professor De Morgan has excellently said,* * It is not for human reason to say what are the simple attributes into which an attribute may be decom- posed.^ And for such a reason it is that I have as &r as possible abstained from treating any terra as known to be simple, 101. Let a term, combined with simple contra- Devdope- ries as alternatives, be called a developement of the ^**^ term as regards the contraries. Thus, AB-j-Ab is called a developement of A as regards B, or in terms of B, or involving B. 102. Any term is same in meaning ajier combi- Continued nation with all the possible combinations of other ^^^^^ ^' terms, and their contraries as alternatives. Since A=AB-i-A5, and, again, A=AC+Ac, we may substitute for A in AB+Aft (§ 51) its expression in terms of C. Thus, A= ABC -h ABc + AbC -h Abe. Again, since A=AD+Ac?, we may substitute a second time, getting A=ABCD4-ABCcZ+ +AJc(^, and so on. * 8yllahu8, p. 60. Digitized by Google .88 PURE LOGIC. Dual term 103« Let any two alternatives, differing only by defined. ^ single part-term and its contrary, be called a diuzl term. Thus, AB + A5 is a dual term as regards B, and ABC-f ABc as regards C, and we may speak of B + J or C + c as the dual part. Seduction 104. A dual term may always he reduced to a %rm^ sm^Ze term by removal of the contrary terms^ mth- out altering the meaning. For the term thus obtained is, by the Law of Duality, the same in meaning as the former dual term (§ 99). Thus, from such a term as AB+Aft, we may always remove the dual part B + ft, and the mean- ing of the term A will still be as before, since A= AB + Ai is a self-evident (§99) truth always in our knowledge. Digitized by Google CONTEART TERMS. 30 CHAPTER X. OF CONTRARY TERMS IN PROPOSITIONS. 105. From any affirmative premise we may infer Affirmative a negative proposition by changing any term on one VJ^ ^^' side only into its contrary. position. From A=B we have Anot=J; for evidently B is not=;J, and hence, by Law of Difference (§77), A=Bnot=ft, or A not = ft. From AB=AC, similarly, AB not=Ac. 106. The two terms of a negative proposition Terms of are contraries. ^^! For the two terms of a negative proposition are tion. different in meaning. Hence there must be some quality or qualities in the meaning of one, and not in that of the other ; thus, the combination of the two terms would mean both the absence and pre- sence of a certain quality or qualities, and would be a contradictory. The two terms then are con- trary (§ 89). 107. A negative proposition may he changed into Negative an affirmative, of which one term is d term of the ^!i-fg " negative^ and the other term this term combined proposi- tcith the contrary of the other term of the negative. ^^* Thus, if A not=B, then A=Aft; or, again, BssaB. For developing A in terms of B (§ 101), we Digitized by Google 40 FUEE LOGIC. have A=AB-hAft, but A and B being contra- ries (§ 106), AB is contradictory or 0. Hence, A=0-fA5=A5(§96). Similarly, we may show B = + «B = aB . So, if ABnot=AC, then AB=ABc. For AB = ABC+ABc=ABc, since ABC is contradictory. And we see that ABc=AB (contrary of AC) =AB (Ac-t-aC-|-ac)=ABc + 0-hO. Since we may now convert any negative propo- sition into an affirmative, it will not be further necessary to use negative propositions in the pro- cess of inference. (§81.) Inference 108. From any contradictory combination we iivTw'opo- '"^y *V^^ ^^^^ ^''^y P^^^ ^f *^^ cotnbination not sitions. itself contradictory is not the same in meaning as the remainder or any greater part. That the two parts differ may be expressed in a negative propo- sition, or its corresponding affirmative. For if the other part be contradictory, it cannot be the same as the first part, which is not contra- dictory. And if neither of the parts is contra- dictory in itself, they cannot be same in meaning, else their combination would not produce a con- tradiction. The affirmative inferences corresponding (§ 107) to the negative ones deduced under this rule may be otherwise had, so that it seems unnecessary to consider the negative inferences further in this place. List of 109. The following are the chief laws or con- ^^^*- ditions of logic : — Digitized by Google LAWS OF LOGIC. 41 Condition or postulate. The meaning of a term must be same throughout any piece of reasoning ; so that A=A, B=B and so on. (§ 14.) Law of Sameness, (§ 25.) A=B=C; hence A=C. Law of Simplicity. (§ 42.) AA=A, BBB=B, and so on! Law of Same Parts and WTioles, (§ 44.) A=B; hence AC=BC. I^w of Unity. (§ 69.) A+A=A, B + Bh-B=B, and so on. Law of Contradiction. (§ 90.) Aa=0, AB5=0, and so on. Law of Duality. (§ 99.) A=A(B + 6)=AB4AJ A=A(B4-*) (C+c) =ABC-f ABc-hA6C-|-Aic and so on. It seems likely that these are the primary and sufficient laws of thought, and others only corol- laries of them. Logic may treat only of known samenesses of things ; and differences of things need be noticed, only for the exclusion from pure logical thought of all that is self-contradictory. In pure number and its science, on the other hand, differences of things only are noticed. The Laws of Simplicity, Unity, Contradiction and Duality furnish the universal premises of reasoning. The Law of Sameness is of altogether a higher order, involving Inference, or the Judg- ment of Judgments. Digitized by Google 42 PURE LOGIC. CHAPTEE XI. OF INBIBECT INFERENCE. Tlseofde- 110. Taken by itself, the developement of a vdopement ^^^^ (^ ^qj^ ^^^ ^ ^^ ^^^^ knowledge about it. But taken with the premises of a problem, we may learn that some of the alternatives of the develope- ment are contradictory and to be rejected. The remaining alternatives then form a new and often useiul expression for the term. Indirect HI. In thus using a developement we are said inference. ^^ ^^^ indirectly y because we use the premise to show what a term is, not directly by the Law of Sameness, but indirectly by showing what it is not. Indirect Inference is direct inference with the aid of self-evident premises derived from the Laws of Contradiction and Duality. But all Inference is stiU by the Law of Sameness. 112. Let A=B : required expressions for A, B, a, b, inferred from this premise. Develope these terms as follows (§ 101) : — A=AB-hA5 B=AB-f-aB rt=aB -\-ab 5=Aft +ab Examine which of the alternatives AB, A6, aB, ab, are contradictory according to the premise A=B. Digitized by Google INDIRECT INFERENCE. 43 A combined with A=B gives A=AB B „ „ „ „ AB=B a „ „ „ „ Aa=aB=0 b „ „ „ „ Aft=B6=0 Hence we learn tHat dB and Ab are contradic- tory, and may be rejected, and that AB is not contradictory according to the premise. Of ab^ which is not foimd among any of the above terms, we can learn nothing from the premise, and it therefore cannot be known to be contradictory. Striking out dB and Ab in the developements of A, B, a, ft, we have — A=AB+0 =AB B=AB-hO =AB a = -f-aft= ab 5=0 +ab=ab m 113. We have here the two inferences A=AB Inferences. B=AB which might have been had from the premise by combination (§ 45), and from which we may pass back by elimination of AB to the premise. We also have a=aft, and ft=saft, which could not have been had by direct inference. And by eli- minating ab between these two we have the new inference a = ft. This result, indeed, that from the eameness of meaning of two terins, we may infer the sameness of meaning of their simple contranea is evidently true. 114. By a similar method we may draw infer- Inference ences from any nxunber of premises, namely, by I^J^J^^ developing any required term in respect of other terms, and striking out the combinations which are shown to be contradictory in any premise. Digitized by Google 44 PURE LOGIC. Method of indirect inference. Develope- '/nent. Compari- son. Included subject. Excluded, Thus, from A=B and B=C, to infer expres- sions for A and a, we develope these terms as follows : — A=ABC+ABc+A5C+AJc a=aBC+ aBc+ abC-habc By combination we then, when possible, render one side of each premise same with each of the alternative combinations, and learn from the other side whether the combination is known to be con- tradictory by the premise. All the combinations in the above developements will be found contra- dictory, except ABC and abcj and we thus get the inferences A=ABC, and a=a6c, of which the former indeed might have been got directly. 115. The process of indirect inference may similarly be applied to drawing any possible infer- ence or expression from any series of premises, however numerous and complicated. The full process may be abbreviated according to the fol- lowing series of rules, which may be said to form THE METHOD OF INDIRECT INFERENCE: — 1. Any premises being given, form a combi- nation containing every term involved therein (§87). Change successively each simple term of this into its contrary, so as to form all the possible combinations of the simple terms and their con- traries. 2. Combine successively each such combina- tion with both members of a premise. When the combination forms a contradiction with neither aide of a premise, call it an included subject of the premise ; when it forms a contradiction with both sides, call it an excluded subject of the premise ; Digitized *ed by Google INDIRECT INFERENCE. 45 when it forms a contradiction with one side only, call it a contradictory combination or subject^ and C&ntradic- etrike it out. ^*^- We may call either an included or excluded subject a possible subject, as distinguished from a Possible. contradictory combination or impossible svhject. Impossible. 3. Perform the same process with each premise. Eepeated Then a combination is an included subject of a '^'^P^^' series of premises, when it is an included subject of any one ; it is a contradictory subject when it is a contradictory of any one ; it is an excluded subject when it is an excluded subject of every premise. 4. The expression for any term involved in the Selection. premises consists of all the included and excluded subjects containing the term, treated as alterna- tives. 5. Such expression may be simplified by re- Reduction, ducing all dual terms (§ 104), and by intrinsic elimination (§ 52) of all terms not required in the expression. 6. When it is observed that the expression Mimina-^ of a term contains a combination which would not ^*^^* occur in the expression of any contrary of that term, we may eliminate the part of the combina- tion common to the term and its expression. (See below, § 117,) 7. Unless each term of the premises and the Contradic- contrary of each appear in one or otlier of the ^^^ . ., / , . ; . . , 1 I premises. possible subjects, the premises must be deemed inconsistent or contradictory. Hence there must always remain at least two possible subjects. (§ 159.) Digitized by Google 46 PURE LOGIC. Example, Bevelope- ment. Compari- son. ABC ABc AbC Abe aBC dBc abC abc Selection. Elimina' turn. Elimina- tion ex- plained. 116. Required by the above process the infer- ences of the premise A=BC. The possible combinations of the terms A,B,C, and their contraries, are as given in the margin. Each of these being combined with both sides of the premise, we have the following results : — ABC =ABC ABC included subject ABc =ABCc =0 ABc contradiction AhQ =AB6C =0 AbC contradiction A^c =AB6Cc=0 0=AaBC=aBC 0=AaBc =aBCc =0 0=AaiC=aBiC =0 0— Aa6c =aB^Cc =0 AbC Abc aBC aBc ahC abc contradiction contradiction excluded subject excluded subject excluded subject It appears, then, that the four combinations ABc to aBC are to be struck out, and only the rest retained as possible subjects. Suppose we now require an expression for the term b as inferred from the premise A=BC. Select from the included and excluded subjects such as contain b, namely abC and abc. Then b=iabC-{-abc, but as aC occurs only with 6, and not with B, its contraiy, we may, by Rule 6, eliminate b from abC ; hence ft=aC+a5c. 117. The validity of this last elimination is seen by drawing the expression for aC, which is abC. Then between bssiabC-\-abc, and abC=aG, we may eliminate abC by substituting (§51) its ex- pression aC. And similarly in all other cases to which the rule applies. We might also reduce the expression for b by Rule 5, as follows : — b=ahC + abc=iah (C-\-c) = ab. Digitized by Google INDIRECT INFERENCE. 47 118. To express a we have Other in- a=aBc + abC + abc, f"^"^^' but observing that none of Be, bC, be, occur with A, so that Bc=aBc, 6C=aftC, ic=aftc, we sub- stitute these simpler terms, eliminating a ; whence asszBc+bC+bCj an evident truth (§ 113). 119. Similarly, we may draw any of the follow- Other in- ing inferences:— ^'^''''''' A=ABC=AB=AC B=AC+aBc C=AB+a6C c=dB'\-abc::^ac aB=Bc aC=6C a6=a^C-t-a5c=a5 (no inference) ac^aBc-j-abc^ac (no inference). 120. Observe that since B and C are samely Relation of related to A, we may get any inference concerning ■" ^^^ ^' one of these terms from the similar inference con- cerning the other by interchanging B and C, ft and c (§ 56). Before proceeding to further examples of in- direct inference, we may make the following observations. 121. When any term appears on both sides of a Excluded premise, as A in AB = AC, any combination con- *^^*^^*- taining its contrary, a, is an excluded subject. Thus, in combining any term with both sides of a proposition, we render any contrary of the term an excluded subject. So, in mathematics we introduce a new root into an equation when we multiply both sides by a ^tor. Digitized by Google 48 PURE LOGIC. Ofinferior import- ance. Plural pi'emises. Identical proposi- tion. Common SUl)J€Ct. 122. An excluded subject, though admitting of inference and admitted into inferences, is of infe- rior and often of no importance. As its name expresses, it is usually a combination concerning which we do not desire knowledge. The sphere of an argument, or the Universe of Thought^ con- tains all the included subjects. An excluded sub- ject is such as lies beyond this sphere or umverse. But we are obliged to consider excluded subjects, because the excluded subject of one premise may be the included subjfect of other premises. 123. When a premise is plural in one or both sides, an excluded subject is a contrary of all the alternatives on both sides, and a contradictory combination is a contrary of all on one side, and not of all on the other side. 124. Of an identical proposition the term itself appearing on either side is the only included sub- ject. All others are excluded, and there are no contradictory combinations. Its useless nature is thus evident. 125. Any subject of a proposition remains an included, excluded, or contradictory subject as before, after combination with any imrelated terms. Thus, if the argument be restricted to a sphere or common subject^ defined by certain terms, these do not need expression in each premise, but may be retained as an exterior condition. Thus, by ABCD ( ) we might mean that ABCD is to be understood as combined with each term of any premises placed within the brackets. ABCD is then the common subject of the premises, which must contain no contrary of this. And any con- Digitized by Google INDIRECT INFERENCE. 49 trary of ABCD is an excluded subject of tlie whole. 126. Any set of terms which always occur in Ofun- the premises in unbroken combination may be combina- treated as a simple term. tions. Thus, if BC occur always thus in combination, we may write for it, say D, and then d or not-BChbC + Bc-^hc. 127. Any set of alternatives which always occur Unbroken together in the premises as alternatives may be P^^^^ treated as a single term. Thus, if B and C occur always as alternatives, we may for B -|- C write, say D, and then d or neither B nor C is be, 128. Any proposition may be treated under the Simple form of A=B, so long as we do not require to ?J^^^' treat its part- terms or alternatives separately. (By §§126,127.) 129. Hence the convenience in every branch of Technical knowledge of using technical terms to stand for every large set of terms which usually occur to- gether. But such terms become the source of error if we do not carefully keep before us their defini- tions, those adopted premises in which we express the set of combined or alternative terms for which we substitute a technical term. 130. In that branch of knowledge, however, Meta- called First Philosojjhy, which is analytic^ and ^J!^ aims at resolving things, or our thoughts about them, into their simplest components, the use of technical terms is ^llacious. Such terms cannot assist analysis, since each arises from the synthesis of many simpler terms, fonmiTTy its definition. £ Digitized by Google 60 PUBE LOGIC. Interrupt- ed process. Unrelated premites. All reasoning, then, in Metaphysics or First Phi- losophy, ought to be carried on in the simplest and most vernacular elements of speech. Ana- lytic science should be like a mill which grinds down the ordinary grains of thought into their smallest and simplest particles. It is in the bake- house we should combine these particles again into loaves of a size and consistency suitable for ordi- nary use. But most metaphysical reasoners, it seems to me, have mistaken the mill and the bakehouse. 131. It is not always necessary to carry out the process of inference exactly as in the rules. Each or any premise may be treated as a separate one, if desirable, and its possible subjects afterwards combined with the possible subjects of other pre- mises. We may thus successively add premises, or try the effect of supposed ones. For instance, since AB and ah are the possible subjects of A=B, and BC and he of B=C, the possible combinations of these, namely ABC and dbc^ are the possible subjects of the two premises combined, observing that AB6c and aJBC are contradictory. 132. If premises be related, the indirect infe- rences will include aU possible direct inferences. From imrelated premises we shall also get such inferences as are possible. Thus, from the unrelated premises \ have A=B, C=D, A=BCD + Bc(Z ei=ABc + aJc, and so on. Digitized by Google INDIRECT INFERENCE, 51 133. It does not seem possible to give any Proof of general proof that the conclusions of the indirect *^^T^^ method must agree with those of the direct me- thod, which will make its truth any the more evident Such proof could be little less than a general recapitulation of the several Laws of Thought. 134. It hardly needs to be pointed out that the EuclicPs method of indirect inference is equivalent to ^^^^^^^^ Euclid's indirect demonstration, or reductio ad tion. ahmrdum. Euclid assumes the developement of alternatives, usually that of equal or greater or 'less, and showing that two of these lead to a con- tradiction, establishes the truth of the third. 136. Nor is this process of reasoning at all new Common or uncommon in any branch of knowledge save ^f«<?/'*w- 1*1.1 1,1 . ^ ^«^f^^ logic, which was supposed to be the science of method. all reasoning. Simple instances occur perhaps as frequently as instances of direct inference, and complicated instances are only rendered scarce by the limited powers of human memory and atten- tion. Among instances of indirect argument we may place all those discourses in which a writer or speaker states several possible alternatives or cases of his subject, and, after showing some of them to be impossible, concludes the rest to be necessary, or else proceeds further to develope and consider these with regard to other premises (§ 131). A good instance is found in Paley's Argument on the Divine Benevolence (Moral Phil., Book II. chap. V.). The old logical process called abscissio infiniti has a close relation to indirect inference. £2 Digitized by Google 52 PURE LOGIC. Quotation, 136. Even brute animals, it would seem, may reason by the indirect method :— * This creature, saith Chrysippus (of the dog), is not void of Logick : for, when in following any beast he cometh to three several ways, he smelleth • to the one, and then to the second ; and if he find that the beast which he pursueth be not fled one of these two ways, he presently, without smelling any further to it, taketh the third way ; which, saith the same Philosopher, is as if he reasoned thus : the Beast must be gone either this, or this, or the other way ; but neither this nor this; ErgOj. the third : so away he runneth.' Sir W. Raleigh, Digitized by Google COMMON LOGIC. 53 CHAPTER XII. OP RELATION TO COMMON LOGIC. Before giving examples of the processes of logical inference as now set forth, it will be well to- consider the relation of our system to the logic of common thought. 137. In ordinary reasoning it will be found that Ordinary there is great economy of thought. Not only are ^^^^^ large collections of attributes and things grouped together under the fewest possible terms, but only those particular attributes of the things under consideration on which the reasoning turns are brought forward. A certain natural disinclination to exertion causes us to simplify our modes of thought as much as possible, and to leave in the background everything that is not essential. Thus when we say man is mortal^ we mean that the at- tributes of mortality are among the attributes of man. But we leave out those infinitely nume- rous attributes of man which are not comprised under mortality, because we do not happen to be occupied with them. The proposition, then, in this form is not that equation of qualities, that statement of perfect sameness or equivalence of meaning, which we have taken as a proposition. 138. It may be objected that we ought to take Digitized by Google 54 PURE LOGIC. Equation or same- " neas the true form of reason- ing. Quantifiea" tion of pre- dicate. the proposition as we find it in common thought. Aristotle so took it, and his system has had a long reign. Some of the expounders of his system even denied that there could be a proposition of two universal and equivalent terms. They could not have committed a greater error or more com- pletely misrepresented the ordinary course of reasoning. Not only, as a fact, do the several sciences establish multitudes of propositions of which the two terms are equivalent and universal, but all definitions are propositions of this kind, and the definitions requisite in connecting the meanings of more and less complex terms, must always form a large part of our data in reasoning. If we iurther consider that even Aristotle's nega- tive propositions have a universal predicate, that men show a constant tendency to treat the predi- cate of the proposition A as universal, whence several common kinds of fallacy, and that reasoning from same to same things may be detected as the fundamental principle of all the sciences, we need have no hesitation in treating the equation as the true proposition, and Aristotie's form as an imperfect proposition. It is thus the Law of Sameness, not the dictum of Aristotie, which governs reasoning. 139, It is only of veiy late years that the im- perfection of the ordinary proposition has been properly pointed out. It is the discovery of the so-called quantification of the predicate which has reduced the proposition to the form of a con- vertible equation, and opened out to logic an indefinite field of improvement. Digitized by Google BOOLIPS SYSTEM. 55 140. Professor Boole's system, first published Bool^s in his ^Mathematical Analysis of LogiCy in '-^^^Sf^* 1847, involves this newly discovered quantifica- tion of the predicate. According to Boole, the some J which is the adjective of particular logical quality, is an indefinite class symhoL Men are some mortals is expressed by him in the equation, x=vy, where x instructs us to select from the universe all things that are men, and y to select all things that are mortal. The proposition then informs us that the things which are men consist of an indefinite selection from among the things which are mortal, v being the symbol of this in- definite quantity or class selected. 141. One more step seems to me necessary. It Further is to separate completely the qualitative and ^^ '^^^" quantitative meanings of all logical terms, in* eluding the word some. In the qualitative form of the proposition man is some mortal — or more correctly speaking, man is some kind of mortal — ^we interpret some or some kind as meaning an in- ' definite and perhaps unknown collection of quali- ties, which being added to the quahties mortalj give the known qualities of num. In the quanti- tative form men are some mortals^ we have the equivalent statement that the collection of indi- viduals in the class som^ mortals is the collection of individuals in the class men, 142. It is strange that the purely qualitative Qtuditaiive form of proposition man is some kind of mortal j F^^^P*^^ which is the most distinct form of statement, and ded, is perhaps the most prevalent, both in science and ordinary thought^ was totally disregarded by Digitized by Google 56 PURE LOGIC. logicians, at least as the foundation of a system of logic. ' The Logicians, until our day,' says Professor De Morgan,* * have considered the ex- tent of a term as the only object of logic, under the name of the logical whole ;' the intent was called by them the metaphysical whole, and was excluded from logic' 'Some' 143. It will be seen that this word some or ^kind/^^^ 50W6 kind, the source of so much difficulty and error, must in our system be treated as a term of indefinite and imknown meaning. It is an unknown term, not only at the beginning of a problem, but throughout it. In no two premises then can the term some or some kind be taken to mean the same set of qualities. Thus we cannot argue through or eliminate a term with some, while at least it retains this unknown term : that is to say, we can never use it as a common term (§ 27) in direct inference. Thus, if A is some B, and some B is some C, we cannot eliminate some B ' getting A is some C, because some being of imknown meaning, the some B is not necessarily the same in both cases. This is still more plain in the form A is some kind of B, and some kind ofB is some kind of C, for it is obvious that the one kind of B is not necessarily the same as the other. 144. Since the term some or some kind is not only unknown but remains unknown throughout any argmnent, we might conveniently appropriate to it some symbol such as U, to remind us of its Special conditions. Thus no term U is to be taken !^, p. 61. Digitized by Google OLD PROPOSITIONS. 57 as same with any other term U, or U=U is not known to be true. But in the propositions A and E it is always open to us, and is best to eli- minate U by writing for it the other member of the proposition (§ 52). Thus, A=UB, meaning that A is some kind of B, involves three terms. It is much better written as A=AB, involving only A and B, and yet perfectly expressing that the qualities of B are among those of A, but not necessarily those of A all among those of B. 146. The four propositions of the old logic may Aristotle's thus find expression in our system : — proposi- A=UB or A=AB A=Uft or A=Aft UA=UB or CA=DB UA=Uft or CA=D6. 146. Two new propositions of De Morgan's DeMor- system are thus expressed : — gansjpro- '' ^ posittons. Everything is either A or B A=b Some things are neither A nor B a =ib. 147. All these propositions, and as many more Thomson^s as may be proposed, can be brought and partially P^'oposi- treated (§ 128) under the form A=B, which I believe to be the simple form of all reasoning. The existence of doubly-imiversal propositions of this kind was far from being unknown to many of the School Logicians, but out of deference to the Aristotelian system, such propositions were neglected. The present Archbishop of York first embodied this proposition in a system of logic, giving it the name U. (Thomson's * Outlines,' passim.) A . Every A is B E . , No AisB I . Some A is B . Some A is not B Digitized by Google 58 PURE LOGIC. CHAPTER XIII. EXAMPLES OF THE METHOD. In this chapter I shall place some miscellaneous examples of inference according to the system of the foregoing chapters, suited to show the power of its method, or its relation to the old logic. . . 148. Let us take a syllogism in Felapton. in Fdap- '' ° ton. No A is B A=U5 = A5 Every A is C A=UC=AC ^ SomeC is not B (§ 145.) Direct From A=Aft we might by combination (§ 45) infer&me, infer AC = AftC, and from A = AC, A6=AftC; whence AC = A6C^A5, or AC = A5, which is a more precise statement of UC = TJb, or some C is not B, the Aristotelian conclusion. We may, however, obtain this conclusion, as well as all other possible ones, by indirect infer- ence. AbC Of the possible combinations of A, B, C, a, b, c, aBC ABC and ABc are contradicted by the first pre- oBc mise, and Abe (as well as ABc) is contradicted by «*C the second premise. A6C, aBC, aBc, abO, ahc, are o-bc the remaining combinations in which we find there is no relation between B and Cper se, since B occurs with C and c, and C occurs with B and b» But Digitized by Google EXAMPLES. 59 AC = A5C, and A& = A&C, whence, by elimina- tion, AC = A6, the same conclusion as before. The following conclusions may also be drawn : a = a (BC -h Be + 5C + 5c) = a (B -h h) (C + c) = a (no inference) B =aB 6 r= AC H- ahO + ahc = AC + a6 C = A5 + aBC -H abO = A6 + aC c = aBc + ahc = ac a5 = ahC H- aJc = aft (C + c) = ah (no inference) aC=aBC + ahC = aC (B + 6) =aC (no inference) he = ahc, 149. The premise AB = CD is of some interest. It contradicts the combinations ABCc?, ABcD, ARcd^ which are AB and not CD, and A6CD, aBCD, aftCD, which are CD and not AB. From the remainder we easily draw the inferences A= BCD -h A5C(Z + A5cD + Aftcc? a=BCc;+BcD +Bc(^ +a6C(^-f aftcD-f-aftce?. Observing that A and B enter samely into the premise, we may easily deduce the expressions for B and h by interchanging A and B, a and h in the above ; thus (§§ 54-56) — B = ACD + dBCd -h aBcD + d&cd. And since A, B enter samely with C, D, we might deduce the corresponding expressions for C, D, and c, a, by interchanging at once A with C, and B with D, or A with D, and B with C. From the expression for a we thus get d = CBa + CftA + Qha + dcRa + dchA + dcha. Observe, that if the expression for A be com- Example : AB-CD ABCD AbCd AbcD Abed oBCd aBcD oBcd abCd aM) abed Digitized by Google 60 PUBE LOGIC. bined with that for a, nothing but contradictory terms will be the result, verifying Aa=0. And, if we combine the expressions for any terms not contrary, as B and d, we get the same result as we might have drawn by the separate applica* tion of the process. Thus, B(Z= AJCD + dBQd + a^cd= + oBd. In expressions thus derived there will often appear, as in the above instance, a superfluous and contradictory term (A5CD, a contrary of Be?), but being only an alternative, the proposition is not untrue. Example : 150. As an example of a premise with a plural ^^^""^ term, let us take A=B-fC. In comparing the eight combinations of A, B, C, ABC a^ Jj c, with the premise, any one is contradictory ^^ which contains A without containing either B or C ; or, again, which contains either B or C without containing A. Thus, ABC, ABc, and A6C, are the included subjects, dbc is an excluded subject, and the rest are contradictory. We may draw the inferences A=BC+Bc4-6C B=:ABC+Ac=AB 6=rA6C +a C=ABC+A5 = AC c=ABc +a. Observe that B and C enter samely, so that their expressions may be mutually derived by interchange. AbC abo Digitized by Google EXAMPLES. 61 161. The premise As=Bc+5C differs from the Example: last in the very important point that A cannot T^ at once be B and C. It has the included subjects ABc and AbCy ABc and the excluded subjects aBC and abc. The A60 following expressions are seen to be simple and ^^^ symmetrical, and it is instructive to form their ^ combinations. A=Bc +hG a=:BC-h6c B=Ac H-aC ft=AC+ac C=AJ +aB c=AB + aJ. 162. From two negative premises we can infer Negative no AristoteHan conclusion (§§ 77, 78). It is weU P^^^^^^ to show that this remains true when the negative , propositions are converted into their corresponding affirmatives (§ 107). Let us take the premises A is not the same as B, C is not the same as B. These may be expressed by the affirmative propositions As=A5, C=6C. If we go through the process of indirect in- AbC ference, and attempt to express A and C in ^-g^ terms of each other, we shall obtain : — abO abc A^AbC+Abc=:Ab (C+c)=Aft C=A6C+a5C=iC (A-ho)=iC. These are the premises over again, and there can be no new inference, except B=aBc. ^ Digitized by Google 62 PURE LOGIC. SoluUons of form Sorites. A=6 o=B a=l> A6 aB ab oB A6 AB AB AB oB ab <tb Aft. Example of two pre- mises. ABCD aBCD oBcD abCD abcD abed 163. The proposition A=B being the simplest form of statement, its foil solution is given below, and the solutions of the similar propositions A= J, a=B, a=5, are inferred by interchanging A and a, B and b. Premise A = B Included subject AB Excluded subject ab Contradiction Ab Contradiction aB 154. Let us take A=ABC, B+C=BD-hCD. We have, by direct inference from the second premise, BC=BCD (§ 45) Hence A=ABC=ABCD (§ 26.) The indirect process gives four included and two excluded subjects, as in the margin. Hence not only the above inference, but the following, among other possible ones :— a=aBCD+aBcD +abCD-\'abcD-\-abcd =aBD +a5D +abd =aD -i-abd =aBD +ab C=ABD +aBCD + a6CD=ABD+aCD c=aBcD -f «JcD ^abcd =acD -^abd Bc = aBcD cD=acD 165. The ordinary Sorites is easily and clearly solved in this system. Taking four premises such as A=AB B=BC C=CD D=DE, many inferences will be evident Digitized by Googk EXAMPLES. 63 from the following series of the subjects, or pos- sible combinations. ABODE aBCDE aftCDE aJcDE ahcM ahcde ^Included subjects. Excluded subjects. 166. The Dilemma of the old logic is easily Dilemma. included in our system, when we supply a term which is suppressed or imderstood in its usual statement. The dilenmia is as follows : — If A is B, E is F, and if C is D, E is F: but, either A is B, or C is D, therefore E is F. Adopting Wallis's reduction to the categorical form, and supplying some term G, to express the present circumstances, or the case in which either A is B, or C is D, we have the pre- mises AB=ABEF CD=CDEF G=ABG+CDG. By the direct process alone we get the required conclusion that, under the condition G, E is F ; thus — GE= (AB -h CD^, GE= ABEFG + CDEFG= GEF or, GE=GEF. 157. The following is known as a Destructive Destructive Ck)nditional Syllogism. Tm^ If A is B, C is D ; but C is not D ; there- fore, A is not B. Digitized by Google 64 PURE LOGIC. Forms of old logic. Complex 'problem. Supplying the suppressed term, say E, express-* ing the circumstances in which A is not B, the following is the statement of this syllogism in our system : — AB=ABCD CE=Ce?E. By direct inference ABE=ABD.CE=ABD.CJE=0. Hence ABE is known to be contradictory ; there- fore (§ 108), AE is not ABE, or in the circum- stances E, A is not B. 168. The forms of the old logic being compre- hended in this system along with an indefinite multitude of other forms, logicians can only pro- perly accept this generalisation, due to Boole, by throwing off as dead encumbrances the useless dis- tinctions of the Aristotelian system. The past history of the Science must not, as hitherto, bar its progress. And Logic will be developed almost like Mathematics, when Logicians like Mathema- ticians discriminate between the Study of Thought and the Study of Antiquarian Lore. I will now give a few complex problems, more suited to show the power of the method. 159. Let the premises be A=B-i-C B=c -^d o=cD AD = BCD. And let it be required to infer the description of any term, say a. By the indirect process, we shall find that the only combination uncontra- dicted by one or other premise is ABCc?. Digitized by Google EXAMPLES. 65 Thus, we find there cannot be any a at all, without contradiction, whatever may be the mean- ing of this result.* It means, doubtless, that the premises are contradictory. (§ 115.7.) * The following law, being of a lees evident character Law of in- than the rest^ has been placed apart finity. Every logical term miist have its contrary. That is to say: — Whatever quality we treat as present we may also treat as absent. There is thus no boundaiy to the universe of logic. Universe of No term can be proposed wide enough to cover its whole ^J^"' ?^' sphere ; for the contrary of any term must add a sphere ^^ of indefinite magnitude. Let U be the universe ; then u\b not included in U. Nor will special terms limit the universe. Thing existing has its contrary in thing not existing. Thing thinkable has its contrary in thing not thinkable. Even thing, the widest noun in the language, has a contrary in that which is not a thing. Of course the above is only true speaking in the strictest logical sense, and using all terms in the most perfect generality. If the above be granted as true, every proposition of the Contradic • form A =B + 6 must be regarded as contradictory of a law torypropo- of thought. For the contrary of A firom the above is **^*^^* B6, a contradiction, or A is used as having no contrary, and forming the universe. Also every system of premises must be rejected which Contradic- altogether contradicts any term or terms. Thus in the ^^ V^^' indirect process we must always have at least two combina- ^**^* tions remaining possible, one of which must contain the contrary of each simple term in the other. In this view the peculiar premises A«B + C B«c +d (§ 159) contain subtle contradictions. For a according to the first premise must be he, and being c, it must by the second premise be B, and hence by first premise also, A, or both A and a, B and b. But this subject needs more consideration. Digitized by Google 66 PURE LOGIC. We also easily infer any of the following : — Problem, ABcDE ABcde AbCDe aBcDE oBcde abCDe abcDB abcDe obcdE abcde Com- plicated A:=BCd B=AC(Z C=ABd d=ABC. AB=Cd AC=B(i BG=Ad ABC = ABGd ABd=ABCd etc. 160. The following premises are such as might easily occur in physical science : — A=ABc +AbC B=BDE +Bde C=CDe. The series of possible combinations in the mar- gin gives by inspection perhaps the most useM information, but the following are a few formal inferences. A= ABcDE + ABc^e + AhCBe BcD=BcDE abd=abcd cd= ABcde +aBcde +abcd(EA-e) =:Bcde+abd bCD -AbCDe -{•abCDe =JCDe. There is no relation between abc and D and E. 161. I conclude with the solution of a still more complicated system of premises. A4-C + E = B+D + F Bc-\-bC=J)e+dE AD=AD/ D=e C=Cd. Digitized by Google EXAMPLES. 67 The possible cqmbinations are : — ABcDe/ aBceZEF ABcc^EF o^cdEf AJ^cdEf aJCc?EF Whence the following, among many other in- ferences, may be drawn : — A=BcDe/-f ABccZE + AftCcffiF Bc=ADe/+Bcc?E D=ABc6/ cc^=BE c=ABcD/ rfE=ABcc^ + JCF+aBc=Bc6?-h6CF JJE=^>CF=WE AF= ABc^F 4- AJCe^EF aF=aBc(ZEF +a^>C(ZEF c?/=Bcc?E/ h =«>C^EF Q=hQdEY=h, Whence the remarkable and unexpected rela- tion C=6, which it would not be easy to detect in the premises. 162. Inferences may be verified by combining Verifica- the expressions of two or more terms, and com- ^*^* paring the result with the expression of the combined term as drawn from the series of pos- sible combinations. For instance, in the problem last given (§ 161), we may c«mbine the expres- sion for A with that for 6?E, as follows : — A . dE=(BcDe/+ ABcc^E 4- A^>Cc^EF)(Bc^-h ^>GF) =04- ABc^E + + + A. JCc^EF, the contradictory combinations being struck out. f2 Digitized by Google 68 PURE LOGIC, But the expressions thus obtamed may not always be in their simplest terms. Sedttction 163. The reduction of inferences to their Mry?*^^" simplest terms, it may be remarked, is in no way essential to their truth; it only renders them more pregnant with information. It is, perhaps, the only part of the process in which there is any difficulty. Worhingof 164. In working these logical problems, it has the process, i)een found very convenient to have a series of combinations of terms beginning with those of A, B, and proceeding up to those of A, B, C, D, E, F, or more, engraved upon a common writing slate. In any given problem, the series is chosen which just furnishes sufficient letters for the dis- tinct terms. The contradictory combinations may then be rapidly struck out, and the remaining combinations lie ready before the eye. Digitized by Google BOOLE'S EXAMPLE. 69 CHAPTER Xiy. COMPARISON WITH BOOLE's SYSTEM. 166. To show the power and facility of this method, as compared with that of Professor Boole, it will be sufficient, as regards those abready ac- quainted with Professor Boole's system, to present the solution of one of his complex examples. Thus, let us follow Professor Boole's investigation of 8emof& Senior's definition of wealth, namely* — that wealth ^^^^ 18 what is transferable, limited in supply , and either productive of pleasure or preventive of pain, (Boole, p. 106.) Let A = Wealth B = Transferable C=Limited in supply D=Productive of pleasure E=Preventive of pain. The definition in question is expressed by the proposition A=BC(DE+D6+<?E) which includes all the combinations of D, E, d, e, except de. * Here, as usually elsewhere, I take words in intent of meaning, and transform most statements accordingly. Digitized by Google '70 PURE LOGIC. Expression forC. ABODE ABCDe ABCdE oBCde flBcDE dBcDe oBcdE oBcde abCDE abCDe obCdE abCde abcDE abcDe abcdE abode Striking out the dual term (E-f e) from BCD (E-f e), we may state the definition in the more concise form A=BCD + BCdE. We may pass over Prof. Boole*s expression for A, after intrinsic elimination of E (A=BCD -f ABCef), as being sufficiently obvious. 166. . Required C in terms of A, B, D (Boole, p. 107). Forming all the possible combinations of A, B, C, D, E, and their contraries, and comparing them with the premise, we shall find all the combina- tions firom ABCde to aBCdE inclusive contra- dicted. The remaining subjects are as in the margin. Selecting the terms containing C, we have C= ABCDE + ABCDe + ABCc?E + aBCde + a5CDE +abCDe -\-ahCdE -{-abCde. Striking out the dual terms (E -|- e), and intrin- sically eliminating remaining E's or e's by substi- tution of C, we have C=ABCT>+ABCd+aBCd+abCD + abCd. Eliminating C fix)m ABCD (§ 117), because ABD=ABCD, and striking out the dual terms (A+a) and (B+d), we have either of the ex- pressions — C=ABD + BCc? +abC C=ABC-haBC^+aJC. From the latter we read, WTiat is limited in supply is either wealthy transferable (and either productive of pleasure or not, ABC), or else some Digitized by Google BOOLE'S. EXAMPLE. 71 .kind of what is not wealthy hut is either not trans- ferable (abC), or, if transferable, is not productive of pleasure {oBCd), This conduaioii is exactly equivalent to that of Professor Boole, on p. 108. 167. His so-called secondary propositions, Negative namely, * 1. Wealth that is intransferable and pro- ^^^^^ ductive of pleasure, does not exist ; ' and * 2. Wealth that is intransferable and not productive of pleasure does not exist,' are negative conclu- sions implied in the striking out of the contra- dictory combinations A5CDE, A5CD«, AftcDE, AftcDe, and AftCe^E, AJbQde, AbcdE, Abcde, which are easily reducible to A5D (C-f c) (E+e)=0 A6D=0 Abd (C + c) (E + e)=0 Abd =0. The expression * does not exist ' is open to ex- ception. 168. Again, required an expression for produc- tive of pleasure (D), in terms of wealth (A), and Expremm preventive of pain (E). (Boole, p. 111.) The complete collection of combinations con- taining D is ABCDE a5CDE ABCDe ahCDe aBcDE a^cDE dQcDe ahcDe. We may then write D as follows : — D=ABCDE+ABCD64-o(Bc+6C+&c)(E4-6)D. But we may observe also that ADE=ABCDE and A6=ABCDe. Digitized by Google 72 PVRE LOGIC. Hence we may substitute ADE and Ae for the two first terms of the expression for D. We may also strike out the dual term (E+c) in the third term, and eliminate the plural term (Bc + ftC + ftc) intrinsically (§ 52) by substitution of D. Thus we get the expression in the required terms : D=ADE+Ae+aD, which may be translated into these words : — Whc^ is productive of happiness is either some kind of wealth preventive of pain, or any hind of wealth not preventive of pain, or some hind of what is not wealth, (Boole, p. Ill ad fin,) Expression 169. For the expression of d we easily select for d. <?=Ac?E+arfe+ac?E=ArfE + «d of which the meaning is — What is not prod\ictive of pleasure is either some hind of wealth preventive of pain, or some hind of what is not wealth. (Boole, p. 112.) Other in' 170. These are the chief inferences furnished ferences. ^^ |^ Boole. From the list of possible combi- nations we could easily add a great many more inferences, in iact as many more, as may be drawn concerning any of the five terms A, B, C, D, E, and their contraries. Thus for CE expressed in the remaining terms, we have CE=ABCDE +ABCdE + obCDE+abCdE ==(ABCE + abCE) (D + d). Striking out the dual term (D-f-rf) and extrin- sicaUy eliminating C in ABCE, since we observe that ABE=ABCE, we have Digitized by Google BOOLirS EXAMPLE. 73 CE=ABE+aJCE which may be translated — What is limited in supply, and preventive of pain, is either wealth, transferable and preventive of pain, or some hind of what is not wealth and not transferable. But ive may oflen find that there is no special relation to express. Thus, in trying to express aftCD in terms of E we find aftCD=aftCD (E + c)=a6CD. 171. Besides affording these formal .deductions, General the series of possible combinations will often give f^^^^ us at a glance a clear and valuable notion of the nations. manner in which the universe of our subject is made up. In this instance we see that for wealth we have the three combinations BCDE, BCDe and BGdE, and that thus for not-wealth (a) we have all possible combinations of B, C, D, E, except those three. With aB we have Cde and € (DE + D6-f-^E+<?e), and with ab, we have all possible combinations of C, D and E. Thus the definition gives no relation between what is not wealth and not transferable, and what is limited in supply, productive of pleasure, or preventive of pairu 172. It is the character of this logical system, Generality in common with that of Professor Boole, that it is ^f ^^ ^*" perfectly general. The same rules which govern the inferences fi'om one or two premises, involving two or three terms, are applicable without the slightest modification to any number of premises. Digitized by Google 74 PURE LOGIC. involving any ntunber of terms. Of course the working of the inferences becomes rapidly more laborioiis as the complexity of the problem increases, and a considerable liability to mistake arises. But this is in the nature of things, and the process of inference, consisting in the mere comparison of terms as to their sameness or difference, seems to me the simplest process that can be conceived. Convpari- 173. Compared with Professor Boole's system, BooI^s ^^ ^^ mathematical dress, this system shows the si/stem. following advantages. 1. Every process is of self-evident nature and force, and governed by laws as simple and primary as those of Euclid's axioms. 2. The process is infallible, and gives no imin- terpretable or anomalous results. 3. The inferences may be drawn with far less labour than in Professor Boole's system, which generally requires a separate computation and de* velopement for each inference. Digitized by Google BOOLirS SYSTEM. 75 CHAPTER XV. REMARKS ON BOOLE's SYSTEM, AND ON THE RELATION OF LOGIC AND MATHEMATICS. 1 74. So long as Professor Boole's system of mathematical logic was capable of giving results beyond the power of any other system, it had in this feet an impregnable strong- hold. Those who were not prepared to draw the same inferences in some other manner could not quarrel with the manner of Professor Boole. But if it be true that the system of the foregoing chapters is of equal power with Professor Boole's system, the case is altered. There are now two systems of notation, giving the same formal results, one of which gives them with self-evident force and mean- ing, the other by dark and symbolic processes. The burden of proof is shifted, and it must be for the author or sup- porters of the dark system to show that it is in some way superior to the evident system. 175. It is not to be denied that Boole's system is con- sistent and perfect within itself. It is, perhaps, one of the most marvellous and admirable pieces of reasoning ever put together. Indeed, if Professor Ferrier, in his Institutes of Metaphysics, is right in holding that the chief excellence of a system is in being reasoned and consistent within itself, then Professor Boole's is nearly or quite the most perfect system ever struck out by a single writer. 176. But a system perfect within itself may not be a perfect representation of the natural system of human thought. The laws and conditions of thought as laid down in the system may not correspond to the laws and condi- tions of thought in reality. If so, the system will not be Digitized by Google 76 REMARKS ON one of Pure and Natural Logic. Such is, I believe, the case. Professor Boole's system is Pure Logic fettered with a condition which converts it from a purely logical into a numerical system. His inferences are not logical inferences ; hence they require to be interpreted, or translated back into logical inferences, which might have been had without ever quitting the self-evident processes of pure logic. Among various objections which I might urge to Boole's system, regarded as purely logical in purpose, are four chief ones to which I shall here confine my attention. First Objection, 177. Boole's symbols are essentially different from the names or symbols of common discourse — his logic is not the logic of common thought. Professor Boole uses the symbol + to join terms together, on the understanding that they are logical contraries, which cannot be predicated of the same thing or combined together witiiout contradiction. He says (p. 32) — * In strictness, the words " and," " or," interposed between the terms descriptive of two or more classes of objects, imply that those classes are quite distinct, so that no member of one is found in another.' 178. This I altogether dispute. In the ordinary use of these conjunctions, we do not necessarily join logical contraries only ; and when terms so joined do prove to be logically contrary, it is by virtue of a tacit premise, some- thing in the meaning of the names and our knowledge of them, which teaches us they are contrary. And when our knowledge of the meanings of tfee words joined is defec- tive, it will often be impossible to decide whether terms joined by conjunctions are contrary or not. 179. Take, for instance, the proposition — * A peer is either a duke, or a marquis, or an earl, or a viscount, or a baron.' If expressed in Professor Boole's symbols, it would be implied that a peer cannot be at once a duke and marquis, or marquis and earl. Yet many peers do possess two or more titles, and the Prince of Wales is Duke of Comyrall, Earl of Chester, Baron Renfrew, &c. K it were enacted Digitized by Google BOOLE'S SYSTEM. 77 by parliament that no peer should have more than one title, this would be the tacit premise which Professor Boole assumes to exist. Again, — * Academic graduates are either bachelors, mas- ters, or doctors ' does not imply that a graduate can be only one of these; the higher degree does not annul the lower. Shakespeare's lines, — * Beauty, truth, and rarity, Grace in all simplicity, Here inclosed in cinders Ke. # * # » To this urn let those repair That are either true or feir,' — certainly do not imply that beauty, truth, rarity, grace, and the true and fair are incompatible notions, so that no instance of one is an instance of another. In the sentence — ' Eepentance is not a single act, but a habit or virtue,' it cannot be implied that a virtue is not a habit ; by Aristotle's definition it is. Milton has the expression in one of his sonnets — * Unstain'd by gold or fee,* where it is obvious that if the fee is not always gold, the gold is a fee or bribe. Tennyson has the expression * wreath or anadem.' Most readers would be quite uncertain whether a wreath may be an anadem, or an anadem a wreath, or whether they are quite distinct or quite the same. From Darwin's * Origin,' I take the expression, * When we see any part or organ developed in a remarkable degree or manner.' In this, or is used twice, and neither time disjunctively. For ii part and organ are not synonjmoua, at any rate an organ is a part. And it is obvious that a part may be developed at the same time both in an extra- ordinary degree and manner, although such cases may be comparatively rare. 180. From a careful examination of ordinary writings, it will be foxmd .that the meanings of terms joined by * and ' * or' vary from absolute identity up to absolute contrariety. There is no logical condition of contrariety at all, and when we do choose contrary expresflions, it is because our Digitized by Google 78 REMARKS ON subject demands it. The matter, not the form of an ex- pression, points out whether terms are exclusive. And if there is one point on which logicians are agreed, it is that logic is formal, and pays no regard to anything not formally expressed. (See § 48.) 181. And if a further proof were wanted that Professor Boole's symbols do not correspond to those of language, we have only to turn to his own work. He actually translates one same sentence into different sets of symbols, according to the view he takes of the matter in hand. For instance, (p. 59) he interprets * Either productive of pleasure or preventive of pain ' so as not to exclude things both pro- ductive of pleasure and preventive of pain. * It is plain,' he remarks, ' from the nature of the subject, that the ex- pression " either productive of pleasure or preventive of pain," in the above definition, is meant to be equivalent to " either productive of pleasure ; or if not productive of pleasure, preventive of pain." ' And in remarking upon other possible interpretations, he says, * that before attempting to translate our data into the rigorous language of symbols, it is above all things necessary to ascertain the intended import of the words we are using.' (p. 60). This simply amounts to consulting the matter, and Professor Boole's symbols thus constantly imply restrictions not expressed in the forms of language, but existing, if at all, as tacit or understood premises. 182. In my system, on the contrary, I take A+B not to imply at all that A may not be B, but if this be the case, it must be owing to an expressed premise A= AJ or A=5. 183. How essential Professor Boole's restriction on his symbols is to the stability of his system, one instance will show. Take his proposition on p. 35 — and give the following meanings to a:, y, z : — a; = Caesar 3/= Conqueror of the Gauls z= First emperor of Rome. Digitized by Google BOOLE'S SYSTEM. 79 Now, there is nothing logically absurd in saying * Gsesar is the conqueror of the Grauls or the first emperor of Rome/ It is quite conceivable that a person should remember just enough of history to make this statement and nothing more. And there is nothing in the logical character of the terms to decide whether the conqueror could or could not be the same person as the first emperor. But now take Professor Boole's inference from the propo- sition aj=y -fz, namely x — -?=y got by subtracting z from either side of x^=y + z. Then we have the strange infer- ence: — CcBsary provided he is not the first emperor of Eomej is the conqueror of the Gauls, This leads me to my second objection to Professor Boole's system. Second Objection, 184. There are no such operations as addition and sub^ traction in pure logic. The operations of logic are the combination and separa- tion of terms, or their meanings, corresponding to multi- plication and division in mathematics. I cannot support this statement without going at once to the gist of the whole matter. 185. Number, then, and the science of number, arise out of logic, and the conditions of number are defined by logic. It has been thought that units are units inasmuch as they are perfectly similar. For instance, three apples are three units, inasmuch as each has exactly the same qualities as the other in being an apple. The truth is exactly opposite to this. Units are units inasmuch as they are logically contrary. In so far as three apples are exactly like each other, one could not be distinguished from the other. Were there three apples, or any three things, so perfectly similar in every way that we could not teU the difference, they would be but one thing, just as, by the law of imity before stated, A+A+A=A. But then we must remember that among the logical characters of a thing is its position in space with relation to other things, not to spe£^ of its position in time. Now, when we speak of Digitized by Google 80 REMARKS ON three apples, we mean three things, which, however per- fectly same they may be in all other qualities, occupy different places, and are therefore distinct things. In so far as they are same they are one ; in that they are dif* ferent they are three, 186. The meaning of an abstract unit is something only known as logically distinct from or contrary to other things. The meaning of a concrete unit is the abstract unit with certain qualities known or defined. For instance, in A (1'4-1" + 1"0=A'+A" + A'" the meaning of the units 1', 1", 1"\ is that each is something logically distinct from the other , and when we predicate of each of these that it is A, say an apple, we get three dis- tinct A's, A' + A'' + A'". So in multiplication, ttoice two is four — (1 + 1)(1 + 1)=1 + 1 + 1 + 1. The logical significance of the process is that if we have two logically distinct notions, and we divide each into two logically distinct notions, we get four logically distinct notions. In logical formulsB (A + o) (B + ^)= AB + Aft -f aB + ah, where A and a, B and 6, express logical contraries. 187. Now addition, subtraction, multiplication, and di- vision, are alike true as modes of reasoning in numbers, where we have the logical condition of a unit as a constant restriction. But addition and subtraction do not exist, and do not give true results, in a system of piure logic, free from the condition of number. For instance, take the logical proposition — A+B+C=A+D+E Meaning what is either A or B or C is either A or D or Ej and vice versa. There being no exterior restrictions of meaning whatever, except that the same term must always have the same meaning (§ 14), we do not know which of A, D, K, is B, nor which is C; nor, conversely, do we know which of A, B, C, is D, nor which is £. The proposition alone gives us no such information. In these circumstances, the action of subtraction does not apply. It is not necessarily true that, if from same (equal) Digitized by Google BOOLE'S SYSTEM. 81 things we take same (equal) things the remainders are same (eqiial). It is not allowable for us to subtract the same thing (A) fix)m both sides of the above proposition, and thence infer — B+C=D+E. This is not true if, for instance, each of B and C is the same as £, and D is the same as A, which has been taken away. Yet the equivalent inference by combination will be valid. We may combine a with both sides of the propo- sition, and we have aA+aB+aC=aA+aD+aE or, striking out the contradictory terms oA, we have aB+aC=oD + «E. 188. But subtraction is valid under the logical restriction that the several alternatives of a term sh^ be mutually exclusive or contrary. Let (1) AMN+BMn+CwN=AMN+DMn+EmN in which it is obviously impossible that AMN can be either DMti or Et/iN, contraries of AMN, or any one of the three alternatives any other. Then we may freely subtract AMN from both sides, getting the necessaiy inference (2) BMw+CwN=DM»+EwN. This subtraction, however, is merely equivalent to the combination with both sides of the proposition (1) of the term (Mw+wN); for the combination being performed, and contradictory terms struck out, it will be found that the proposition (2) results. 189. In short, when alternatives are contraries of each other, subtraction of one is. exactly equivalent to combina- tion with the rest. The axiom (Boole, p. 36), that * if equal things are taken from equal things, the remainders are equal,' is nothing but a case of the Law of Combination (§ 44), that if same (equal) terms be combined with same (equal) terms, the wholes are same (equal). G Digitized by Google 82 SEMARKS ON Take the self-eyident proposition AB+Aft+aB + a&=AB-fAft+aB+a5 Any terms, say aB + aft, may be subtracted from both sides by combining the other terms AB+Aft with each side of the proposition. Then (AB+AJ) (AB + Aft+aB + a^) =(AB+ A^)(AB + Aft + aB+aft) AB+AJ+0 . . +0=AB+Aft+0-|- . . 0+0. And what is true of this self-evident case must be true when the premise is not self-evident. 190. Having thus established our liberty to subtract same terms, provided all alternatives are contraries, we have the corresponding liberty to add by the inverse process. 191. The processes of addition and subtraction thus arise out of the logical process of combination. The axioms of addition and subtraction are only valid under a logical con- dition, which is certainly not applicable to thought or language generally. And this condition is that which logic imposes upon number, that each two imits shall be contrary logical alternatives. It is logic which reduces to a imit, by the Law of Unity, A+A=A, any two alternatives known to be the same, so that the science of number treating of units, treats of alternatives known to be different or con- trary. But logic itself is the superior science^ and may treat of alternatives of which it is not known whether they are same or different, 192. It is the self-evident logical Law of Unity, then, which lays the foundations of number. This law merely amounts to sajring that a thing cannot and must not be distinguished from itself. We commit an error against this law, when in counting over coins, for instance, to ascertain their numbers, that is, how many logically distinct coins there are, we count the self-same coin two or more times, making the coins for instance C'+C"+C"+C'"+C""+ instead of C' + C" + C'" + C''"-|- It is by the Law of Unity that C" + C"=C", or the same coin counted Digitized by Google BOOLE'S SYSTEM. 83 twice is but one coin in number. In this case no attention is paid to differences of time ; but in many cases, things otherwise perfectly the same, like the beats of a pendulum, are distinguished and made into different units by one being before or after the other in time;. Third Objection. 193. My third objection to Professor Boole's system is, that it is inconsistent with the self-evident law of thoughty the Law of Unity, (A+A=A.) Prof. Boole having assumed as a condition of his system that each two terms must be logically distinct, is imable to recognise the Law of Unity. It is contradictory of the basis of his system. The term a;, in his system, means all things with the qvMity x, denoting the things in extent, while connoting die quality in intent. If by 1 we denote all things of every quality, and then subtract, as in num- bers, all those things which have the quality a?, the re- mainder must consist of all things of the quality not-x. Thus, a?+(l — x) means in his system all x's with all not-x^s, which, taken together, must make up all things, or 1. But let us now attempt by multiplication with x, to select all x's from this expression for all things, x(x+l'-x)=:x+x^x. Professor Boole would here cross out one +x against one — a?, leaving one -fa?, the required expression for all x^s. It is surely self-evident, however, that x+x is equivalent to X alone, whether we regard it in extent of meaning, as all the x's added to all the x's, which is simply all the x^s, or in intent of meaning, as either x or x, which is surely x. Thus, x+x—x is really 0, and not a:, the required result, and it is apparent that the process of subtraction in logic is inconsistent with the self-evident Law of Unity. 194. It is probable, indeed, that Professor Boole would altogether reflise to recognise such an expression as x+x—x, on the ground that it does not obey the condition of his symbols that each two alternatives shall be distinct and contrary, x+x not being so. It may be answered, g2 Digitized by Googk 84 . BEMARKS ON that the expression has been arrived at by operations enounced as universally valid, which ought to give true results. And if it be simply said that x-\'X'^x is not interpretdble in Professor Boole*s system, it may be again answered, that when .translated into its equivalent in words, the expression x-^-x—x has a very plain meaning. It is * either x qr x, provided it be not a?,' and this, I must hold, . is simply not x, although it ought to be Xy according to the mode in which it was got. 195. In founding his system, Boole assumed that there cannot be two terms A + B, the same in meaning or names of the same thing ; the laws of thought require nothing of the kind, and cannot require it, because among known and unknown terms, any two such as A+B may prove to be names of the same thing AB. Thought merely reduces the meaning of two same terms AB -j- AB, by the Law of Unity, to be the same as that of one term AB. And when it is once known that all terms in question are contraries of each other, or naturally exclusive and distinct, then Boole's system and the whole science of numbers apply. 196. It is on this account that my objections have no bearing against Professor Boole's system as applied to the Calculus of Probabilities, so far as I can understand the subject. For it is a high advantage to that calculus to have to treat only events mutually exclusive, probabilities being then capable of simple addition and subtraction.* It seems likely, indeed, that this distinction of exclusive 'and unexclusive alternatives is the Gordian knot in which all the abstract logical sciences meet and are entangled. Fourth Objection, 197. The last objection that I shall at present urge against Professor Boole's system is, that the symbols -J^, §,^, ^, establish for themselves no logical meaning, and only bear a meaning derived from some method of reasoning not con-' tained in the symbolic system. The meanings, in short, are those reached in the self-evident indirect method of the present work. * See De Morgan's Syllabus, § 243. Digitized by Google BOOLE'S SYSTEM. 198. Professor Boole expreesljr allows, as regards one of these symbols at least, ^, that it is not his method which gives any meaning to the symbol. It is the peculiarity of his system, that he bestows a meaning on his symbols by interpretation. The interpretation of ^ is explained on pp. 89, 90, and he says, ' Although the above determina- Jiion of the significance of the symbol -§■ is founded upon the examination of a particular case, yet the principle in- volved in the demonstration is general, and diere are no circumstances under which the symbol can present itself to which the same mode of analysis is inapplicable.' Again (p. 91), ' Its actual interpretation, however, as an ind.efinite dass symbol, cannot, I conceive, except upon the ground of ansiogy, be deduced from its aridimetical properties, but must be established experimentally.' 199. If I understand this aright, it simply means, that wherever a term appears in a conclusion with the' symbol ^ afiixed, we may, by a mode of analysis, by some process of pure reasoning apart from the symbolic process from which ^ emerged, ascertain that the meaning of ^ is some, an indefinite class term. The symbol ^ is unknown until we give it a meaning. Before, therefore, we can know what meaning to give, and be sure that this meaning is right, it seems to me we must have another distinct and intuitive system by which to get that meaning. Professor Boole's system, then, as regards the symbol ^, is not the system bestowing certain knowledge ; it is, at most, a sys- tem pointing out truths which, by another intuitive system of reasoning, we may know to be certainly true. 200. It is sufficient to show this with regard to a single symbol %, because the incapacity of a system, even in a single instance, proves the necessity for another system to support it. I believe that the other symbols, \y ^, ^, are open to exactly the same remarks, but from the way in which Mr. Boole treats them, involving the whole condi- tions of his system, it would be a lengthy matter to explaii^. 201. The obscure symbols -J-, ii h \i ^*v® *^® follow- ing correspondence with the forms of the present system. \ appearing as the coefficient of a term means that the term is an included subject of the premise, so that, if Digitized by Google 86 BEMARKS ON combined with both members of the premise, it produces a self-contradictory term With neither side (§ 115). Similarly, % means that the term is an excluded subject of the premise j producing a self- contradictory with both sides of the premise. . And either ^ or f means that the term is a aelf-covtra- dietary or impossible term, producing a self-contradictory, term with one side only of the premise. 208. The correspondence of these obscure forms with the self-evident inferences of the present system is so close and obvious, as to suggest irresistibly that Professor Boole's operations with his abstract calculus of 1 and 0, are a mere counterpart of self-evident operations with the intelligible symbols of pure logic. Professor Boole starts from logical notions, and self-evident laws of thought; he suddenly transmutes his formulae into obscure mathematical counter- parts, and after various intricate manoeuvres, arrives at certain forms, corresponding to forms arrived at directly and intuitively by ordinary or Pure Logic — ^by that analysis, from which the interpretation of his symbols was reached and proved. And by this interpretation he transfers the meaning and force of pure logical conclusions to obscure forms, which, if they have meaning, have certainly no de- monstrative force of themselves. Boole's system is like the shadow, the ghost, the reflected image of logic, seen among the derivatives of logic. 803. Supposing it prove true that Professor Boole's Cal- ciQuB of 1 and has no real logical force and meaning, it cannot be denied that there is still something highly remarkable^ something highly mysterious in the &ct, that logical forms can be turned into numeral forms, and while treated as numbers, still possess formal logical truth. It proves that diere is a certain identity of logical and numerical reasoning. Logic and mathematics are certainly not inde- pendent. And the clue to their connection seems to consist in distinct logical terms forming the units of mathematics. 804< Things as they appear to us in the reality of nature, are clothed in inexhaustible attributes, set as it were in a frame of time and space. By our mental powers we ab- stract first time, then space, and then attribute ailer attri- Digitized by Google BOOLE'S SYSTEM. 87 bute, until we can finally think of things as abstract units deprived of all attributes, and only retaining the original logical condition of things, that each is distinct from others. In logic we argue upon things as same and one, in number we reason upon them as distinct and many. 206. Supposing it be ultimately allowed that Professor Boole's calculus of 1 and is not really logic at all ; that his system is foimded upon one condition, that of exclusive terms, which does not belong to thought in general, but only numerical thought; and that it ignored one law of logic, the Law of Unity, which really distinguishes a logical from a numerical system — ^these errors scarcely detract from the beauty and originality of the views he laid open. Logic, after his work, is to logic before his work, as mathe- matics with equations of any degree are to mathematics with equations of one or two degrees. He generalised logic so that it became possible to obtain any true inference from premises of any degree of complexity, and the work I have attempted has been little more than to translate his forms into processes of self-evident meaning and force. Owens College, ]\1a.nchestei% : November, 1863. LONDON PBIKTBD BT BDWABD. STANFORD CHABING GBOSS, S.W. Digitized by Google BY THE SAME. JPoH %vo. with Two Diatoms, price ^. doth^ A SEBIOUS FALL IN THE VALUE OF GOLD ASOEBTAINED, and its Social Effects Set Forth. Frice 8#. 6d. eaek^ coloured^ DIAGBAM SHEWING ALL THE WEEELT ACCOUNTS OF THE BANE OF ENGLAND, sinoe the Paseing of the Bank Act of 1844, with the Amount of Bank of England, Priyate, and Joint-Stock Bank Promissory Notes, in Circulation during each Week, and the Bank Minimum Bate of Discount. DIAGRAM SHEWING THE PRICE OF THE ENGLISH FUNDS, the Price of Wheat, the Number of Bank- ruptcies, and the Bate of Discount, Monthly, since 1781 ; so fiir as the same haye been ascertained. *•* The aboYS Diagrams represent to the eye all the useftd results of tables containing 125,000 figures, carefully compiled for the purposes of the Diagrams, Digitized by Google Digitized by Google i Digitized by'